2.26.7 Problem 7

Internal problem ID [13643]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 1, section 1.4. Equations Containing Polynomial Functions of y. subsection 1.4.1-2 Abel equations of the first kind.
Problem number : 7
Date solved : Monday, January 05, 2026 at 01:47:27 AM
CAS classification : [[_homogeneous, `class G`], _Abel]

2.26.7.1 Solved using first_order_ode_exact

2.197 (sec)

Entering first order ode exact solver

\begin{align*} y^{\prime }&=a x y^{3}+y^{2} b \\ \end{align*}

To solve an ode of the form

\begin{equation} M\left ( x,y\right ) +N\left ( x,y\right ) \frac {dy}{dx}=0\tag {A}\end{equation}

We assume there exists a function \(\phi \left ( x,y\right ) =c\) where \(c\) is constant, that satisfies the ode. Taking derivative of \(\phi \) w.r.t. \(x\) gives

\[ \frac {d}{dx}\phi \left ( x,y\right ) =0 \]

Hence

\begin{equation} \frac {\partial \phi }{\partial x}+\frac {\partial \phi }{\partial y}\frac {dy}{dx}=0\tag {B}\end{equation}

Comparing (A,B) shows that

\begin{align*} \frac {\partial \phi }{\partial x} & =M\\ \frac {\partial \phi }{\partial y} & =N \end{align*}

But since \(\frac {\partial ^{2}\phi }{\partial x\partial y}=\frac {\partial ^{2}\phi }{\partial y\partial x}\) then for the above to be valid, we require that

\[ \frac {\partial M}{\partial y}=\frac {\partial N}{\partial x}\]

If the above condition is satisfied, then the original ode is called exact. We still need to determine \(\phi \left ( x,y\right ) \) but at least we know now that we can do that since the condition \(\frac {\partial ^{2}\phi }{\partial x\partial y}=\frac {\partial ^{2}\phi }{\partial y\partial x}\) is satisfied. If this condition is not satisfied then this method will not work and we have to now look for an integrating factor to force this condition, which might or might not exist. The first step is to write the ODE in standard form to check for exactness, which is

\[ M(x,y) \mathop {\mathrm {d}x}+ N(x,y) \mathop {\mathrm {d}y}=0 \tag {1A} \]

Therefore

\begin{align*} \mathop {\mathrm {d}y} &= \left (a x \,y^{3}+y^{2} b\right )\mathop {\mathrm {d}x}\\ \left (-a x \,y^{3}-y^{2} b\right ) \mathop {\mathrm {d}x} + \mathop {\mathrm {d}y} &= 0 \tag {2A} \end{align*}

Comparing (1A) and (2A) shows that

\begin{align*} M(x,y) &= -a x \,y^{3}-y^{2} b\\ N(x,y) &= 1 \end{align*}

The next step is to determine if the ODE is is exact or not. The ODE is exact when the following condition is satisfied

\[ \frac {\partial M}{\partial y} = \frac {\partial N}{\partial x} \]

Using result found above gives

\begin{align*} \frac {\partial M}{\partial y} &= \frac {\partial }{\partial y} \left (-a x \,y^{3}-y^{2} b\right )\\ &= -3 a x \,y^{2}-2 b y \end{align*}

And

\begin{align*} \frac {\partial N}{\partial x} &= \frac {\partial }{\partial x} \left (1\right )\\ &= 0 \end{align*}

Since \(\frac {\partial M}{\partial y} \neq \frac {\partial N}{\partial x}\), then the ODE is not exact. Since the ODE is not exact, we will try to find an integrating factor to make it exact. Let

\begin{align*} A &= \frac {1}{N} \left (\frac {\partial M}{\partial y} - \frac {\partial N}{\partial x} \right ) \\ &=1\left ( \left ( -3 a x \,y^{2}-2 b y\right ) - \left (0 \right ) \right ) \\ &=-3 a x \,y^{2}-2 b y \end{align*}

Since \(A\) depends on \(y\), it can not be used to obtain an integrating factor. We will now try a second method to find an integrating factor. Let

\begin{align*} B &= \frac {1}{M} \left ( \frac {\partial N}{\partial x} - \frac {\partial M}{\partial y} \right ) \\ &=-\frac {1}{y^{2} \left (y a x +b \right )}\left ( \left ( 0\right ) - \left (-3 a x \,y^{2}-2 b y \right ) \right ) \\ &=\frac {-3 y a x -2 b}{y \left (y a x +b \right )} \end{align*}

Since \(B\) depends on \(x\), it can not be used to obtain an integrating factor.We will now try a third method to find an integrating factor. Let

\[ R = \frac { \frac {\partial N}{\partial x} - \frac {\partial M}{\partial y} } {x M - y N} \]

\(R\) is now checked to see if it is a function of only \(t=x y\). Therefore

\begin{align*} R &= \frac { \frac {\partial N}{\partial x} - \frac {\partial M}{\partial y} } {x M - y N} \\ &= \frac {\left (0\right )-\left (-3 a x \,y^{2}-2 b y\right )} {x\left (-a x \,y^{3}-y^{2} b\right ) - y\left (1\right )} \\ &= \frac {-3 y a x -2 b}{x^{2} a \,y^{2}+b x y +1} \end{align*}

Replacing all powers of terms \(x y\) by \(t\) gives

\[ R = \frac {-3 a t -2 b}{a \,t^{2}+b t +1} \]

Since \(R\) depends on \(t\) only, then it can be used to find an integrating factor. Let the integrating factor be \(\mu \) then

\begin{align*} \mu &= e^{\int R \mathop {\mathrm {d}t}} \\ &= e^{\int \left (\frac {-3 a t -2 b}{a \,t^{2}+b t +1}\right )\mathop {\mathrm {d}t} } \end{align*}

The result of integrating gives

\begin{align*} \mu &= e^{-\frac {3 \ln \left (a \,t^{2}+b t +1\right )}{2}+\frac {b \,\operatorname {arctanh}\left (\frac {2 a t +b}{\sqrt {b^{2}-4 a}}\right )}{\sqrt {b^{2}-4 a}} } \\ &= \frac {{\mathrm e}^{\frac {b \,\operatorname {arctanh}\left (\frac {2 a t +b}{\sqrt {b^{2}-4 a}}\right )}{\sqrt {b^{2}-4 a}}}}{\left (a \,t^{2}+b t +1\right )^{{3}/{2}}} \end{align*}

Now \(t\) is replaced back with \(x y\) giving

\[ \mu =\frac {{\mathrm e}^{\frac {b \,\operatorname {arctanh}\left (\frac {2 y a x +b}{\sqrt {b^{2}-4 a}}\right )}{\sqrt {b^{2}-4 a}}}}{\left (x^{2} a \,y^{2}+b x y +1\right )^{{3}/{2}}} \]

Multiplying \(M\) and \(N\) by this integrating factor gives new \(M\) and new \(N\) which are called \( \overline {M}\) and \( \overline {N}\) so not to confuse them with the original \(M\) and \(N\)

\begin{align*} \overline {M} &=\mu M \\ &= \frac {{\mathrm e}^{\frac {b \,\operatorname {arctanh}\left (\frac {2 y a x +b}{\sqrt {b^{2}-4 a}}\right )}{\sqrt {b^{2}-4 a}}}}{\left (x^{2} a \,y^{2}+b x y +1\right )^{{3}/{2}}}\left (-a x \,y^{3}-y^{2} b\right ) \\ &= \frac {\left (-y a x -b \right ) y^{2} {\mathrm e}^{\frac {b \,\operatorname {arctanh}\left (\frac {2 y a x +b}{\sqrt {b^{2}-4 a}}\right )}{\sqrt {b^{2}-4 a}}}}{\left (x^{2} a \,y^{2}+b x y +1\right )^{{3}/{2}}} \end{align*}

And

\begin{align*} \overline {N} &=\mu N \\ &= \frac {{\mathrm e}^{\frac {b \,\operatorname {arctanh}\left (\frac {2 y a x +b}{\sqrt {b^{2}-4 a}}\right )}{\sqrt {b^{2}-4 a}}}}{\left (x^{2} a \,y^{2}+b x y +1\right )^{{3}/{2}}}\left (1\right ) \\ &= \frac {{\mathrm e}^{\frac {b \,\operatorname {arctanh}\left (\frac {2 y a x +b}{\sqrt {b^{2}-4 a}}\right )}{\sqrt {b^{2}-4 a}}}}{\left (x^{2} a \,y^{2}+b x y +1\right )^{{3}/{2}}} \end{align*}

A modified ODE is now obtained from the original ODE, which is exact and can solved. The modified ODE is

\begin{align*} \overline {M} + \overline {N} \frac { \mathop {\mathrm {d}y}}{\mathop {\mathrm {d}x}} &= 0 \\ \left (\frac {\left (-y a x -b \right ) y^{2} {\mathrm e}^{\frac {b \,\operatorname {arctanh}\left (\frac {2 y a x +b}{\sqrt {b^{2}-4 a}}\right )}{\sqrt {b^{2}-4 a}}}}{\left (x^{2} a \,y^{2}+b x y +1\right )^{{3}/{2}}}\right ) + \left (\frac {{\mathrm e}^{\frac {b \,\operatorname {arctanh}\left (\frac {2 y a x +b}{\sqrt {b^{2}-4 a}}\right )}{\sqrt {b^{2}-4 a}}}}{\left (x^{2} a \,y^{2}+b x y +1\right )^{{3}/{2}}}\right ) \frac { \mathop {\mathrm {d}y}}{\mathop {\mathrm {d}x}} &= 0 \end{align*}

The following equations are now set up to solve for the function \(\phi \left (x,y\right )\)

\begin{align*} \frac {\partial \phi }{\partial x } &= \overline {M}\tag {1} \\ \frac {\partial \phi }{\partial y } &= \overline {N}\tag {2} \end{align*}

Integrating (1) w.r.t. \(x\) gives

\begin{align*} \int \frac {\partial \phi }{\partial x} \mathop {\mathrm {d}x} &= \int \overline {M}\mathop {\mathrm {d}x} \\ \int \frac {\partial \phi }{\partial x} \mathop {\mathrm {d}x} &= \int \frac {\left (-y a x -b \right ) y^{2} {\mathrm e}^{\frac {b \,\operatorname {arctanh}\left (\frac {2 y a x +b}{\sqrt {b^{2}-4 a}}\right )}{\sqrt {b^{2}-4 a}}}}{\left (x^{2} a \,y^{2}+b x y +1\right )^{{3}/{2}}}\mathop {\mathrm {d}x} \\ \tag{3} \phi &= \frac {{\mathrm e}^{\frac {b \,\operatorname {arctanh}\left (\frac {2 y a x +b}{\sqrt {b^{2}-4 a}}\right )}{\sqrt {b^{2}-4 a}}} y}{\sqrt {x^{2} a \,y^{2}+b x y +1}}+ f(y) \\ \end{align*}

Where \(f(y)\) is used for the constant of integration since \(\phi \) is a function of both \(x\) and \(y\). Taking derivative of equation (3) w.r.t \(y\) gives

\begin{align*} \tag{4} \frac {\partial \phi }{\partial y} &= \frac {2 b a x \,{\mathrm e}^{\frac {b \,\operatorname {arctanh}\left (\frac {2 y a x +b}{\sqrt {b^{2}-4 a}}\right )}{\sqrt {b^{2}-4 a}}} y}{\left (b^{2}-4 a \right ) \left (-\frac {\left (2 y a x +b \right )^{2}}{b^{2}-4 a}+1\right ) \sqrt {x^{2} a \,y^{2}+b x y +1}}+\frac {{\mathrm e}^{\frac {b \,\operatorname {arctanh}\left (\frac {2 y a x +b}{\sqrt {b^{2}-4 a}}\right )}{\sqrt {b^{2}-4 a}}}}{\sqrt {x^{2} a \,y^{2}+b x y +1}}-\frac {{\mathrm e}^{\frac {b \,\operatorname {arctanh}\left (\frac {2 y a x +b}{\sqrt {b^{2}-4 a}}\right )}{\sqrt {b^{2}-4 a}}} y \left (2 y a \,x^{2}+b x \right )}{2 \left (x^{2} a \,y^{2}+b x y +1\right )^{{3}/{2}}}+f'(y) \\ &=\frac {{\mathrm e}^{\frac {b \,\operatorname {arctanh}\left (\frac {2 y a x +b}{\sqrt {b^{2}-4 a}}\right )}{\sqrt {b^{2}-4 a}}}}{\left (x^{2} a \,y^{2}+b x y +1\right )^{{3}/{2}}}+f'(y) \\ \end{align*}

But equation (2) says that \(\frac {\partial \phi }{\partial y} = \frac {{\mathrm e}^{\frac {b \,\operatorname {arctanh}\left (\frac {2 y a x +b}{\sqrt {b^{2}-4 a}}\right )}{\sqrt {b^{2}-4 a}}}}{\left (x^{2} a \,y^{2}+b x y +1\right )^{{3}/{2}}}\). Therefore equation (4) becomes

\begin{equation} \tag{5} \frac {{\mathrm e}^{\frac {b \,\operatorname {arctanh}\left (\frac {2 y a x +b}{\sqrt {b^{2}-4 a}}\right )}{\sqrt {b^{2}-4 a}}}}{\left (x^{2} a \,y^{2}+b x y +1\right )^{{3}/{2}}} = \frac {{\mathrm e}^{\frac {b \,\operatorname {arctanh}\left (\frac {2 y a x +b}{\sqrt {b^{2}-4 a}}\right )}{\sqrt {b^{2}-4 a}}}}{\left (x^{2} a \,y^{2}+b x y +1\right )^{{3}/{2}}}+f'(y) \end{equation}

Solving equation (5) for \( f'(y)\) gives

\[ f'(y) = 0 \]

Therefore

\[ f(y) = c_1 \]

Where \(c_1\) is constant of integration. Substituting this result for \(f(y)\) into equation (3) gives \(\phi \)

\[ \phi = \frac {{\mathrm e}^{\frac {b \,\operatorname {arctanh}\left (\frac {2 y a x +b}{\sqrt {b^{2}-4 a}}\right )}{\sqrt {b^{2}-4 a}}} y}{\sqrt {x^{2} a \,y^{2}+b x y +1}}+ c_1 \]

But since \(\phi \) itself is a constant function, then let \(\phi =c_2\) where \(c_2\) is new constant and combining \(c_1\) and \(c_2\) constants into the constant \(c_1\) gives the solution as

\[ c_1 = \frac {{\mathrm e}^{\frac {b \,\operatorname {arctanh}\left (\frac {2 y a x +b}{\sqrt {b^{2}-4 a}}\right )}{\sqrt {b^{2}-4 a}}} y}{\sqrt {x^{2} a \,y^{2}+b x y +1}} \]

Summary of solutions found

\begin{align*} \frac {{\mathrm e}^{\frac {b \,\operatorname {arctanh}\left (\frac {2 y a x +b}{\sqrt {b^{2}-4 a}}\right )}{\sqrt {b^{2}-4 a}}} y}{\sqrt {x^{2} a y^{2}+b x y+1}} &= c_1 \\ \end{align*}

2.26.7.2 Solved using first_order_ode_isobaric

1.625 (sec)

Entering first order ode isobaric solver

\begin{align*} y^{\prime }&=a x y^{3}+y^{2} b \\ \end{align*}

Solving for \(y'\) gives

\[ y' = a x y^{3}+y^{2} b \]

An ode \(y^{\prime }=f(x,y)\) is isobaric if

\[ f(t x, t^m y) = t^{m-1} f(x,y)\tag {1} \]

Where here

\[ f(x,y) = a x y^{3}+y^{2} b\tag {2} \]

\(m\) is the order of isobaric. Substituting (2) into (1) and solving for \(m\) gives

\[ m = -1 \]

Since the ode is isobaric of order \(m=-1\), then the substitution

\begin{align*} y&=u x^m \\ &=\frac {u}{x} \end{align*}

Converts the ODE to a separable in \(u \left (x \right )\). Performing this substitution gives

\[ -\frac {u \left (x \right )}{x^{2}}+\frac {u^{\prime }\left (x \right )}{x} = \frac {a u \left (x \right )^{3}}{x^{2}}+\frac {u \left (x \right )^{2} b}{x^{2}} \]

2.26.7.3 Solved using first_order_ode_separable

1.480 (sec)

Entering first order ode separable solver

\begin{align*} -\frac {u \left (x \right )}{x^{2}}+\frac {u^{\prime }\left (x \right )}{x}&=\frac {a u \left (x \right )^{3}}{x^{2}}+\frac {u \left (x \right )^{2} b}{x^{2}} \\ \end{align*}

The ode

\begin{equation} u^{\prime }\left (x \right ) = \frac {u \left (x \right ) \left (a u \left (x \right )^{2}+b u \left (x \right )+1\right )}{x} \end{equation}

is separable as it can be written as

\begin{align*} u^{\prime }\left (x \right )&= \frac {u \left (x \right ) \left (a u \left (x \right )^{2}+b u \left (x \right )+1\right )}{x}\\ &= f(x) g(u) \end{align*}

Where

\begin{align*} f(x) &= \frac {1}{x}\\ g(u) &= u \left (a \,u^{2}+b u +1\right ) \end{align*}

Integrating gives

\begin{align*} \int { \frac {1}{g(u)} \,du} &= \int { f(x) \,dx} \\ \int { \frac {1}{u \left (a \,u^{2}+b u +1\right )}\,du} &= \int { \frac {1}{x} \,dx} \\ \end{align*}

\[ -\frac {\ln \left (a u \left (x \right )^{2}+b u \left (x \right )+1\right )}{2}+\frac {b \,\operatorname {arctanh}\left (\frac {2 a u \left (x \right )+b}{\sqrt {b^{2}-4 a}}\right )}{\sqrt {b^{2}-4 a}}+\ln \left (u \left (x \right )\right )=\ln \left (x \right )+c_1 \]

We now need to find the singular solutions, these are found by finding for what values \(g(u)\) is zero, since we had to divide by this above. Solving \(g(u)=0\) or

\[ u \left (a \,u^{2}+b u +1\right )=0 \]

for \(u \left (x \right )\) gives

\begin{align*} u \left (x \right )&=0\\ u \left (x \right )&=\frac {-b +\sqrt {b^{2}-4 a}}{2 a} \end{align*}

Now we go over each such singular solution and check if it verifies the ode itself and any initial conditions given. If it does not then the singular solution will not be used.

Therefore the solutions found are

\begin{align*} -\frac {\ln \left (a u \left (x \right )^{2}+b u \left (x \right )+1\right )}{2}+\frac {b \,\operatorname {arctanh}\left (\frac {2 a u \left (x \right )+b}{\sqrt {b^{2}-4 a}}\right )}{\sqrt {b^{2}-4 a}}+\ln \left (u \left (x \right )\right ) &= \ln \left (x \right )+c_1 \\ u \left (x \right ) &= 0 \\ u \left (x \right ) &= \frac {-b +\sqrt {b^{2}-4 a}}{2 a} \\ \end{align*}

Summary of solutions found

\begin{align*} -\frac {\ln \left (a u \left (x \right )^{2}+b u \left (x \right )+1\right )}{2}+\frac {b \,\operatorname {arctanh}\left (\frac {2 a u \left (x \right )+b}{\sqrt {b^{2}-4 a}}\right )}{\sqrt {b^{2}-4 a}}+\ln \left (u \left (x \right )\right ) &= \ln \left (x \right )+c_1 \\ u \left (x \right ) &= 0 \\ u \left (x \right ) &= \frac {-b +\sqrt {b^{2}-4 a}}{2 a} \\ \end{align*}

Converting \(-\frac {\ln \left (a u \left (x \right )^{2}+b u \left (x \right )+1\right )}{2}+\frac {b \,\operatorname {arctanh}\left (\frac {2 a u \left (x \right )+b}{\sqrt {b^{2}-4 a}}\right )}{\sqrt {b^{2}-4 a}}+\ln \left (u \left (x \right )\right ) = \ln \left (x \right )+c_1\) back to \(y\) gives

\begin{align*} -\frac {\ln \left (x^{2} a y^{2}+b x y+1\right )}{2}+\frac {b \,\operatorname {arctanh}\left (\frac {2 y a x +b}{\sqrt {b^{2}-4 a}}\right )}{\sqrt {b^{2}-4 a}}+\ln \left (x y\right ) = \ln \left (x \right )+c_1 \end{align*}

Converting \(u \left (x \right ) = 0\) back to \(y\) gives

\begin{align*} x y = 0 \end{align*}

Converting \(u \left (x \right ) = \frac {-b +\sqrt {b^{2}-4 a}}{2 a}\) back to \(y\) gives

\begin{align*} x y = \frac {-b +\sqrt {b^{2}-4 a}}{2 a} \end{align*}

Solving for \(y\) gives

\begin{align*} -\frac {\ln \left (x^{2} a y^{2}+b x y+1\right )}{2}+\frac {b \,\operatorname {arctanh}\left (\frac {2 y a x +b}{\sqrt {b^{2}-4 a}}\right )}{\sqrt {b^{2}-4 a}}+\ln \left (x y\right ) &= \ln \left (x \right )+c_1 \\ y &= 0 \\ y &= \frac {-b +\sqrt {b^{2}-4 a}}{2 a x} \\ \end{align*}

Summary of solutions found

\begin{align*} -\frac {\ln \left (x^{2} a y^{2}+b x y+1\right )}{2}+\frac {b \,\operatorname {arctanh}\left (\frac {2 y a x +b}{\sqrt {b^{2}-4 a}}\right )}{\sqrt {b^{2}-4 a}}+\ln \left (x y\right ) &= \ln \left (x \right )+c_1 \\ y &= 0 \\ y &= \frac {-b +\sqrt {b^{2}-4 a}}{2 a x} \\ \end{align*}

2.26.7.4 Solved using first_order_ode_homog_type_G

1.191 (sec)

Entering first order ode homog type G solver

\begin{align*} y^{\prime }&=a x y^{3}+y^{2} b \\ \end{align*}

Multiplying the right side of the ode, which is \(a x \,y^{3}+y^{2} b\) by \(\frac {x}{y}\) gives

\begin{align*} y^{\prime } &= \left (\frac {x}{y}\right ) a x \,y^{3}+y^{2} b\\ &= x y \left (y a x +b \right )\\ &= F(x,y) \end{align*}

Since \(F \left (x , y\right )\) has \(y\), then let

\begin{align*} f_x&= x \left (\frac {\partial }{\partial x}F \left (x , y\right )\right )\\ &= y x \left (2 y a x +b \right )\\ f_y&= y \left (\frac {\partial }{\partial y}F \left (x , y\right )\right )\\ &= y x \left (2 y a x +b \right )\\ \alpha &= \frac {f_x}{f_y} \\ &=1 \end{align*}

Since \(\alpha \) is independent of \(x,y\) then this is Homogeneous type G.

Let

\begin{align*} y&=\frac {z}{x^ \alpha }\\ &=\frac {z}{x} \end{align*}

Substituting the above back into \(F(x,y)\) gives

\begin{align*} F \left (z \right ) &=z \left (a z +b \right ) \end{align*}

We see that \(F \left (z \right )\) does not depend on \(x\) nor on \(y\). If this was not the case, then this method will not work.

Therefore, the implicit solution is given by

\begin{align*} \ln \left (x \right )- c_1 - \int ^{y x^\alpha } \frac {1}{z \left (\alpha + F(z)\right ) } \,dz & = 0 \end{align*}

Which gives

\[ \ln \left (x \right )-c_1 +\int _{}^{x y}\frac {1}{z \left (-1-z \left (a z +b \right )\right )}d z = 0 \]

The value of the above is

\[ \ln \left (x \right )-c_1 +\frac {\ln \left (x^{2} a y^{2}+b x y+1\right ) \sqrt {b^{2}-4 a}-2 \ln \left (x y\right ) \sqrt {b^{2}-4 a}-2 b \,\operatorname {arctanh}\left (\frac {2 y a x +b}{\sqrt {b^{2}-4 a}}\right )}{2 \sqrt {b^{2}-4 a}} = 0 \]

Simplifying the above gives

\begin{align*} \frac {-2 b \,\operatorname {arctanh}\left (\frac {2 y a x +b}{\sqrt {b^{2}-4 a}}\right )+\left (\ln \left (x^{2} a y^{2}+b x y+1\right )-2 \ln \left (x y\right )+2 \ln \left (x \right )-2 c_1 \right ) \sqrt {b^{2}-4 a}}{2 \sqrt {b^{2}-4 a}} &= 0 \\ \end{align*}

Summary of solutions found

\begin{align*} \frac {-2 b \,\operatorname {arctanh}\left (\frac {2 y a x +b}{\sqrt {b^{2}-4 a}}\right )+\left (\ln \left (x^{2} a y^{2}+b x y+1\right )-2 \ln \left (x y\right )+2 \ln \left (x \right )-2 c_1 \right ) \sqrt {b^{2}-4 a}}{2 \sqrt {b^{2}-4 a}} &= 0 \\ \end{align*}

2.26.7.5 Solved using first_order_ode_abel_first_kind

4.067 (sec)

Entering first order ode abel first kind solver

\begin{align*} y^{\prime }&=a x y^{3}+y^{2} b \\ \end{align*}

This is Abel first kind ODE, it has the form

\[ y^{\prime }= f_0(x)+f_1(x) y +f_2(x)y^{2}+f_3(x)y^{3} \]

Comparing the above to given ODE which is

\begin{align*}y^{\prime }&=a x y^{3}+y^{2} b\tag {1} \end{align*}

Therefore

\begin{align*} f_0 &= 0\\ f_1 &= 0\\ f_2 &= b\\ f_3 &= a x \end{align*}

Hence

\begin{align*} f'_{0} &= 0\\ f'_{3} &= a \end{align*}

Since \(f_2(x)=b\) is not zero, then the followingtransformation is used to remove \(f_2\). Let \(y = u(x) - \frac {f_2}{3 f_3}\) or

\begin{align*} y &= u(x) - \left ( \frac {b}{3 a x} \right ) \\ &= u \left (x \right )-\frac {b}{3 a x} \end{align*}

The above transformation applied to (1) gives a new ODE as

\begin{align*} u^{\prime }\left (x \right ) = a x u \left (x \right )^{3}-\frac {u \left (x \right ) b^{2}}{3 a x}+\frac {2 b^{3}-9 a b}{27 a^{2} x^{2}}\tag {2} \end{align*}

The above ODE (2) can now be solved.

Entering first order ode exact solver To solve an ode of the form

\begin{equation} M\left ( x,y\right ) +N\left ( x,y\right ) \frac {dy}{dx}=0\tag {A}\end{equation}

We assume there exists a function \(\phi \left ( x,y\right ) =c\) where \(c\) is constant, that satisfies the ode. Taking derivative of \(\phi \) w.r.t. \(x\) gives

\[ \frac {d}{dx}\phi \left ( x,y\right ) =0 \]

Hence

\begin{equation} \frac {\partial \phi }{\partial x}+\frac {\partial \phi }{\partial y}\frac {dy}{dx}=0\tag {B}\end{equation}

Comparing (A,B) shows that

\begin{align*} \frac {\partial \phi }{\partial x} & =M\\ \frac {\partial \phi }{\partial y} & =N \end{align*}

But since \(\frac {\partial ^{2}\phi }{\partial x\partial y}=\frac {\partial ^{2}\phi }{\partial y\partial x}\) then for the above to be valid, we require that

\[ \frac {\partial M}{\partial y}=\frac {\partial N}{\partial x}\]

If the above condition is satisfied, then the original ode is called exact. We still need to determine \(\phi \left ( x,y\right ) \) but at least we know now that we can do that since the condition \(\frac {\partial ^{2}\phi }{\partial x\partial y}=\frac {\partial ^{2}\phi }{\partial y\partial x}\) is satisfied. If this condition is not satisfied then this method will not work and we have to now look for an integrating factor to force this condition, which might or might not exist. The first step is to write the ODE in standard form to check for exactness, which is

\[ M(x,u) \mathop {\mathrm {d}x}+ N(x,u) \mathop {\mathrm {d}u}=0 \tag {1A} \]

Therefore

\begin{align*} \mathop {\mathrm {d}u} &= \left (a x \,u^{3}-\frac {u \,b^{2}}{3 a x}+\frac {2 b^{3}-9 a b}{27 a^{2} x^{2}}\right )\mathop {\mathrm {d}x}\\ \left (-a x \,u^{3}+\frac {u \,b^{2}}{3 a x}-\frac {2 b^{3}-9 a b}{27 a^{2} x^{2}}\right ) \mathop {\mathrm {d}x} + \mathop {\mathrm {d}u} &= 0 \tag {2A} \end{align*}

Comparing (1A) and (2A) shows that

\begin{align*} M(x,u) &= -a x \,u^{3}+\frac {u \,b^{2}}{3 a x}-\frac {2 b^{3}-9 a b}{27 a^{2} x^{2}}\\ N(x,u) &= 1 \end{align*}

The next step is to determine if the ODE is is exact or not. The ODE is exact when the following condition is satisfied

\[ \frac {\partial M}{\partial u} = \frac {\partial N}{\partial x} \]

Using result found above gives

\begin{align*} \frac {\partial M}{\partial u} &= \frac {\partial }{\partial u} \left (-a x \,u^{3}+\frac {u \,b^{2}}{3 a x}-\frac {2 b^{3}-9 a b}{27 a^{2} x^{2}}\right )\\ &= -3 a x \,u^{2}+\frac {b^{2}}{3 a x} \end{align*}

And

\begin{align*} \frac {\partial N}{\partial x} &= \frac {\partial }{\partial x} \left (1\right )\\ &= 0 \end{align*}

Since \(\frac {\partial M}{\partial u} \neq \frac {\partial N}{\partial x}\), then the ODE is not exact. Since the ODE is not exact, we will try to find an integrating factor to make it exact. Let

\begin{align*} A &= \frac {1}{N} \left (\frac {\partial M}{\partial u} - \frac {\partial N}{\partial x} \right ) \\ &=1\left ( \left ( -3 a x \,u^{2}+\frac {b^{2}}{3 a x}\right ) - \left (0 \right ) \right ) \\ &=-3 a x \,u^{2}+\frac {b^{2}}{3 a x} \end{align*}

Since \(A\) depends on \(u\), it can not be used to obtain an integrating factor. We will now try a second method to find an integrating factor. Let

\begin{align*} B &= \frac {1}{M} \left ( \frac {\partial N}{\partial x} - \frac {\partial M}{\partial u} \right ) \\ &=-\frac {27 x^{2} a^{2}}{27 a^{3} x^{3} u^{3}-9 \left (b^{2} u x +b \right ) a +2 b^{3}}\left ( \left ( 0\right ) - \left (-3 a x \,u^{2}+\frac {b^{2}}{3 a x} \right ) \right ) \\ &=\frac {-81 a^{3} u^{2} x^{3}+9 a \,b^{2} x}{27 a^{3} x^{3} u^{3}+\left (-9 b^{2} u x -9 b \right ) a +2 b^{3}} \end{align*}

Since \(B\) depends on \(x\), it can not be used to obtain an integrating factor.We will now try a third method to find an integrating factor. Let

\[ R = \frac { \frac {\partial N}{\partial x} - \frac {\partial M}{\partial u} } {x M - y N} \]

\(R\) is now checked to see if it is a function of only \(t=x u\). Therefore

\begin{align*} R &= \frac { \frac {\partial N}{\partial x} - \frac {\partial M}{\partial u} } {x M - y N} \\ &= \frac {\left (0\right )-\left (-3 a x \,u^{2}+\frac {b^{2}}{3 a x}\right )} {x\left (-a x \,u^{3}+\frac {u \,b^{2}}{3 a x}-\frac {2 b^{3}-9 a b}{27 a^{2} x^{2}}\right ) - u\left (1\right )} \\ &= -\frac {9 \left (3 a u x +b \right ) a}{9 a^{2} x^{2} u^{2}+3 a b u x -2 b^{2}+9 a} \end{align*}

Replacing all powers of terms \(x u\) by \(t\) gives

\[ R = -\frac {9 \left (3 a t +b \right ) a}{9 a^{2} t^{2}+3 a b t -2 b^{2}+9 a} \]

Since \(R\) depends on \(t\) only, then it can be used to find an integrating factor. Let the integrating factor be \(\mu \) then

\begin{align*} \mu &= e^{\int R \mathop {\mathrm {d}t}} \\ &= e^{\int \left (-\frac {9 \left (3 a t +b \right ) a}{9 a^{2} t^{2}+3 a b t -2 b^{2}+9 a}\right )\mathop {\mathrm {d}t} } \end{align*}

The result of integrating gives

\begin{align*} \mu &= e^{-\frac {3 \ln \left (9 a^{2} t^{2}+3 a b t -2 b^{2}+9 a \right )}{2}+\frac {a b \,\operatorname {arctanh}\left (\frac {18 a^{2} t +3 a b}{9 \sqrt {a^{2} b^{2}-4 a^{3}}}\right )}{\sqrt {a^{2} b^{2}-4 a^{3}}} } \\ &= \frac {{\mathrm e}^{\frac {a b \,\operatorname {arctanh}\left (\frac {a \left (6 a t +b \right )}{3 \sqrt {a^{2} b^{2}-4 a^{3}}}\right )}{\sqrt {a^{2} b^{2}-4 a^{3}}}}}{\left (9 a^{2} t^{2}+3 a b t -2 b^{2}+9 a \right )^{{3}/{2}}} \end{align*}

Now \(t\) is replaced back with \(x u\) giving

\[ \mu =\frac {{\mathrm e}^{\frac {a b \,\operatorname {arctanh}\left (\frac {a \left (6 a u x +b \right )}{3 \sqrt {a^{2} b^{2}-4 a^{3}}}\right )}{\sqrt {a^{2} b^{2}-4 a^{3}}}}}{\left (9 a^{2} x^{2} u^{2}+3 a b u x -2 b^{2}+9 a \right )^{{3}/{2}}} \]

Multiplying \(M\) and \(N\) by this integrating factor gives new \(M\) and new \(N\) which are called \( \overline {M}\) and \( \overline {N}\) so not to confuse them with the original \(M\) and \(N\)

\begin{align*} \overline {M} &=\mu M \\ &= \frac {{\mathrm e}^{\frac {a b \,\operatorname {arctanh}\left (\frac {a \left (6 a u x +b \right )}{3 \sqrt {a^{2} b^{2}-4 a^{3}}}\right )}{\sqrt {a^{2} b^{2}-4 a^{3}}}}}{\left (9 a^{2} x^{2} u^{2}+3 a b u x -2 b^{2}+9 a \right )^{{3}/{2}}}\left (-a x \,u^{3}+\frac {u \,b^{2}}{3 a x}-\frac {2 b^{3}-9 a b}{27 a^{2} x^{2}}\right ) \\ &= \frac {\left (-27 a^{3} x^{3} u^{3}+9 \left (b^{2} u x +b \right ) a -2 b^{3}\right ) {\mathrm e}^{\frac {a b \,\operatorname {arctanh}\left (\frac {a \left (6 a u x +b \right )}{3 \sqrt {a^{2} b^{2}-4 a^{3}}}\right )}{\sqrt {a^{2} b^{2}-4 a^{3}}}}}{27 \left (9 a^{2} x^{2} u^{2}+3 a b u x -2 b^{2}+9 a \right )^{{3}/{2}} x^{2} a^{2}} \end{align*}

And

\begin{align*} \overline {N} &=\mu N \\ &= \frac {{\mathrm e}^{\frac {a b \,\operatorname {arctanh}\left (\frac {a \left (6 a u x +b \right )}{3 \sqrt {a^{2} b^{2}-4 a^{3}}}\right )}{\sqrt {a^{2} b^{2}-4 a^{3}}}}}{\left (9 a^{2} x^{2} u^{2}+3 a b u x -2 b^{2}+9 a \right )^{{3}/{2}}}\left (1\right ) \\ &= \frac {{\mathrm e}^{\frac {a b \,\operatorname {arctanh}\left (\frac {a \left (6 a u x +b \right )}{3 \sqrt {a^{2} b^{2}-4 a^{3}}}\right )}{\sqrt {a^{2} b^{2}-4 a^{3}}}}}{\left (9 a^{2} x^{2} u^{2}+3 a b u x -2 b^{2}+9 a \right )^{{3}/{2}}} \end{align*}

A modified ODE is now obtained from the original ODE, which is exact and can solved. The modified ODE is

\begin{align*} \overline {M} + \overline {N} \frac { \mathop {\mathrm {d}u}}{\mathop {\mathrm {d}x}} &= 0 \\ \left (\frac {\left (-27 a^{3} x^{3} u^{3}+9 \left (b^{2} u x +b \right ) a -2 b^{3}\right ) {\mathrm e}^{\frac {a b \,\operatorname {arctanh}\left (\frac {a \left (6 a u x +b \right )}{3 \sqrt {a^{2} b^{2}-4 a^{3}}}\right )}{\sqrt {a^{2} b^{2}-4 a^{3}}}}}{27 \left (9 a^{2} x^{2} u^{2}+3 a b u x -2 b^{2}+9 a \right )^{{3}/{2}} x^{2} a^{2}}\right ) + \left (\frac {{\mathrm e}^{\frac {a b \,\operatorname {arctanh}\left (\frac {a \left (6 a u x +b \right )}{3 \sqrt {a^{2} b^{2}-4 a^{3}}}\right )}{\sqrt {a^{2} b^{2}-4 a^{3}}}}}{\left (9 a^{2} x^{2} u^{2}+3 a b u x -2 b^{2}+9 a \right )^{{3}/{2}}}\right ) \frac { \mathop {\mathrm {d}u}}{\mathop {\mathrm {d}x}} &= 0 \end{align*}

The following equations are now set up to solve for the function \(\phi \left (x,u\right )\)

\begin{align*} \frac {\partial \phi }{\partial x } &= \overline {M}\tag {1} \\ \frac {\partial \phi }{\partial u } &= \overline {N}\tag {2} \end{align*}

Integrating (1) w.r.t. \(x\) gives

\begin{align*} \int \frac {\partial \phi }{\partial x} \mathop {\mathrm {d}x} &= \int \overline {M}\mathop {\mathrm {d}x} \\ \int \frac {\partial \phi }{\partial x} \mathop {\mathrm {d}x} &= \int \frac {\left (-27 a^{3} x^{3} u^{3}+9 \left (b^{2} u x +b \right ) a -2 b^{3}\right ) {\mathrm e}^{\frac {a b \,\operatorname {arctanh}\left (\frac {a \left (6 a u x +b \right )}{3 \sqrt {a^{2} b^{2}-4 a^{3}}}\right )}{\sqrt {a^{2} b^{2}-4 a^{3}}}}}{27 \left (9 a^{2} x^{2} u^{2}+3 a b u x -2 b^{2}+9 a \right )^{{3}/{2}} x^{2} a^{2}}\mathop {\mathrm {d}x} \\ \tag{3} \phi &= \frac {{\mathrm e}^{\frac {a b \,\operatorname {arctanh}\left (\frac {a \left (6 a u x +b \right )}{3 \sqrt {a^{2} b^{2}-4 a^{3}}}\right )}{\sqrt {a^{2} b^{2}-4 a^{3}}}} \left (3 a u x -b \right )}{27 a^{2} x \sqrt {9 a^{2} x^{2} u^{2}+3 a b u x -2 b^{2}+9 a}}+ f(u) \\ \end{align*}

Where \(f(u)\) is used for the constant of integration since \(\phi \) is a function of both \(x\) and \(u\). Taking derivative of equation (3) w.r.t \(u\) gives

\begin{align*} \tag{4} \frac {\partial \phi }{\partial u} &= \frac {2 a b \,{\mathrm e}^{\frac {a b \,\operatorname {arctanh}\left (\frac {a \left (6 a u x +b \right )}{3 \sqrt {a^{2} b^{2}-4 a^{3}}}\right )}{\sqrt {a^{2} b^{2}-4 a^{3}}}} \left (3 a u x -b \right )}{27 \left (a^{2} b^{2}-4 a^{3}\right ) \left (-\frac {a^{2} \left (6 a u x +b \right )^{2}}{9 \left (a^{2} b^{2}-4 a^{3}\right )}+1\right ) \sqrt {9 a^{2} x^{2} u^{2}+3 a b u x -2 b^{2}+9 a}}+\frac {{\mathrm e}^{\frac {a b \,\operatorname {arctanh}\left (\frac {a \left (6 a u x +b \right )}{3 \sqrt {a^{2} b^{2}-4 a^{3}}}\right )}{\sqrt {a^{2} b^{2}-4 a^{3}}}}}{9 a \sqrt {9 a^{2} x^{2} u^{2}+3 a b u x -2 b^{2}+9 a}}-\frac {{\mathrm e}^{\frac {a b \,\operatorname {arctanh}\left (\frac {a \left (6 a u x +b \right )}{3 \sqrt {a^{2} b^{2}-4 a^{3}}}\right )}{\sqrt {a^{2} b^{2}-4 a^{3}}}} \left (3 a u x -b \right ) \left (18 u \,a^{2} x^{2}+3 a b x \right )}{54 a^{2} x \left (9 a^{2} x^{2} u^{2}+3 a b u x -2 b^{2}+9 a \right )^{{3}/{2}}}+f'(u) \\ &=\frac {{\mathrm e}^{\frac {a b \,\operatorname {arctanh}\left (\frac {a \left (6 a u x +b \right )}{3 \sqrt {a^{2} b^{2}-4 a^{3}}}\right )}{\sqrt {a^{2} b^{2}-4 a^{3}}}}}{\left (9 a^{2} x^{2} u^{2}+3 a b u x -2 b^{2}+9 a \right )^{{3}/{2}}}+f'(u) \\ \end{align*}

But equation (2) says that \(\frac {\partial \phi }{\partial u} = \frac {{\mathrm e}^{\frac {a b \,\operatorname {arctanh}\left (\frac {a \left (6 a u x +b \right )}{3 \sqrt {a^{2} b^{2}-4 a^{3}}}\right )}{\sqrt {a^{2} b^{2}-4 a^{3}}}}}{\left (9 a^{2} x^{2} u^{2}+3 a b u x -2 b^{2}+9 a \right )^{{3}/{2}}}\). Therefore equation (4) becomes

\begin{equation} \tag{5} \frac {{\mathrm e}^{\frac {a b \,\operatorname {arctanh}\left (\frac {a \left (6 a u x +b \right )}{3 \sqrt {a^{2} b^{2}-4 a^{3}}}\right )}{\sqrt {a^{2} b^{2}-4 a^{3}}}}}{\left (9 a^{2} x^{2} u^{2}+3 a b u x -2 b^{2}+9 a \right )^{{3}/{2}}} = \frac {{\mathrm e}^{\frac {a b \,\operatorname {arctanh}\left (\frac {a \left (6 a u x +b \right )}{3 \sqrt {a^{2} b^{2}-4 a^{3}}}\right )}{\sqrt {a^{2} b^{2}-4 a^{3}}}}}{\left (9 a^{2} x^{2} u^{2}+3 a b u x -2 b^{2}+9 a \right )^{{3}/{2}}}+f'(u) \end{equation}

Solving equation (5) for \( f'(u)\) gives

\[ f'(u) = 0 \]

Therefore

\[ f(u) = c_1 \]

Where \(c_1\) is constant of integration. Substituting this result for \(f(u)\) into equation (3) gives \(\phi \)

\[ \phi = \frac {{\mathrm e}^{\frac {a b \,\operatorname {arctanh}\left (\frac {a \left (6 a u x +b \right )}{3 \sqrt {a^{2} b^{2}-4 a^{3}}}\right )}{\sqrt {a^{2} b^{2}-4 a^{3}}}} \left (3 a u x -b \right )}{27 a^{2} x \sqrt {9 a^{2} x^{2} u^{2}+3 a b u x -2 b^{2}+9 a}}+ c_1 \]

But since \(\phi \) itself is a constant function, then let \(\phi =c_2\) where \(c_2\) is new constant and combining \(c_1\) and \(c_2\) constants into the constant \(c_1\) gives the solution as

\[ c_1 = \frac {{\mathrm e}^{\frac {a b \,\operatorname {arctanh}\left (\frac {a \left (6 a u x +b \right )}{3 \sqrt {a^{2} b^{2}-4 a^{3}}}\right )}{\sqrt {a^{2} b^{2}-4 a^{3}}}} \left (3 a u x -b \right )}{27 a^{2} x \sqrt {9 a^{2} x^{2} u^{2}+3 a b u x -2 b^{2}+9 a}} \]

Substituting \(u=y+\frac {b}{3 a x}\) in the above solution gives

\begin{align*} \frac {{\mathrm e}^{\frac {a b \,\operatorname {arctanh}\left (\frac {a \left (6 \left (y+\frac {b}{3 a x}\right ) a x +b \right )}{3 \sqrt {a^{2} b^{2}-4 a^{3}}}\right )}{\sqrt {a^{2} b^{2}-4 a^{3}}}} \left (3 \left (y+\frac {b}{3 a x}\right ) a x -b \right )}{27 a^{2} x \sqrt {9 a^{2} x^{2} \left (y+\frac {b}{3 a x}\right )^{2}+3 a b \left (y+\frac {b}{3 a x}\right ) x -2 b^{2}+9 a}} = c_1 \end{align*}

Simplifying the above gives

\begin{align*} \frac {{\mathrm e}^{\frac {a b \,\operatorname {arctanh}\left (\frac {a \left (2 y a x +b \right )}{\sqrt {a^{2} b^{2}-4 a^{3}}}\right )}{\sqrt {a^{2} b^{2}-4 a^{3}}}} y}{27 a \sqrt {a \left (x^{2} a y^{2}+b x y+1\right )}} &= c_1 \\ \end{align*}

Summary of solutions found

\begin{align*} \frac {{\mathrm e}^{\frac {a b \,\operatorname {arctanh}\left (\frac {a \left (2 y a x +b \right )}{\sqrt {a^{2} b^{2}-4 a^{3}}}\right )}{\sqrt {a^{2} b^{2}-4 a^{3}}}} y}{27 a \sqrt {a \left (x^{2} a y^{2}+b x y+1\right )}} &= c_1 \\ \end{align*}

2.26.7.6 Solved using first_order_ode_LIE

4.487 (sec)

Entering first order ode LIE solver

\begin{align*} y^{\prime }&=a x y^{3}+y^{2} b \\ \end{align*}

Writing the ode as

\begin{align*} y^{\prime }&=a x \,y^{3}+y^{2} b\\ y^{\prime }&= \omega \left ( x,y\right ) \end{align*}

The condition of Lie symmetry is the linearized PDE given by

\begin{align*} \eta _{x}+\omega \left ( \eta _{y}-\xi _{x}\right ) -\omega ^{2}\xi _{y}-\omega _{x}\xi -\omega _{y}\eta =0\tag {A} \end{align*}

To determine \(\xi ,\eta \) then (A) is solved using ansatz. Making bivariate polynomials of degree 1 to use as anstaz gives

\begin{align*} \tag{1E} \xi &= x a_{2}+y a_{3}+a_{1} \\ \tag{2E} \eta &= x b_{2}+y b_{3}+b_{1} \\ \end{align*}

Where the unknown coefficients are

\[ \{a_{1}, a_{2}, a_{3}, b_{1}, b_{2}, b_{3}\} \]

Substituting equations (1E,2E) and \(\omega \) into (A) gives

\begin{equation} \tag{5E} b_{2}+\left (a x \,y^{3}+y^{2} b \right ) \left (b_{3}-a_{2}\right )-\left (a x \,y^{3}+y^{2} b \right )^{2} a_{3}-a \,y^{3} \left (x a_{2}+y a_{3}+a_{1}\right )-\left (3 a x \,y^{2}+2 b y \right ) \left (x b_{2}+y b_{3}+b_{1}\right ) = 0 \end{equation}

Putting the above in normal form gives

\[ -a^{2} x^{2} y^{6} a_{3}-2 a b x \,y^{5} a_{3}-b^{2} y^{4} a_{3}-3 a \,x^{2} y^{2} b_{2}-2 a x \,y^{3} a_{2}-2 a x \,y^{3} b_{3}-a \,y^{4} a_{3}-3 a x \,y^{2} b_{1}-a \,y^{3} a_{1}-2 b x y b_{2}-b \,y^{2} a_{2}-b \,y^{2} b_{3}-2 b y b_{1}+b_{2} = 0 \]

Setting the numerator to zero gives

\begin{equation} \tag{6E} -a^{2} x^{2} y^{6} a_{3}-2 a b x \,y^{5} a_{3}-b^{2} y^{4} a_{3}-3 a \,x^{2} y^{2} b_{2}-2 a x \,y^{3} a_{2}-2 a x \,y^{3} b_{3}-a \,y^{4} a_{3}-3 a x \,y^{2} b_{1}-a \,y^{3} a_{1}-2 b x y b_{2}-b \,y^{2} a_{2}-b \,y^{2} b_{3}-2 b y b_{1}+b_{2} = 0 \end{equation}

Looking at the above PDE shows the following are all the terms with \(\{x, y\}\) in them.

\[ \{x, y\} \]

The following substitution is now made to be able to collect on all terms with \(\{x, y\}\) in them

\[ \{x = v_{1}, y = v_{2}\} \]

The above PDE (6E) now becomes

\begin{equation} \tag{7E} -a^{2} a_{3} v_{1}^{2} v_{2}^{6}-2 a b a_{3} v_{1} v_{2}^{5}-b^{2} a_{3} v_{2}^{4}-2 a a_{2} v_{1} v_{2}^{3}-a a_{3} v_{2}^{4}-3 a b_{2} v_{1}^{2} v_{2}^{2}-2 a b_{3} v_{1} v_{2}^{3}-a a_{1} v_{2}^{3}-3 a b_{1} v_{1} v_{2}^{2}-b a_{2} v_{2}^{2}-2 b b_{2} v_{1} v_{2}-b b_{3} v_{2}^{2}-2 b b_{1} v_{2}+b_{2} = 0 \end{equation}

Collecting the above on the terms \(v_i\) introduced, and these are

\[ \{v_{1}, v_{2}\} \]

Equation (7E) now becomes

\begin{equation} \tag{8E} -a^{2} a_{3} v_{1}^{2} v_{2}^{6}-3 a b_{2} v_{1}^{2} v_{2}^{2}-2 a b a_{3} v_{1} v_{2}^{5}+\left (-2 a a_{2}-2 a b_{3}\right ) v_{1} v_{2}^{3}-3 a b_{1} v_{1} v_{2}^{2}-2 b b_{2} v_{1} v_{2}+\left (-b^{2} a_{3}-a a_{3}\right ) v_{2}^{4}-a a_{1} v_{2}^{3}+\left (-b a_{2}-b b_{3}\right ) v_{2}^{2}-2 b b_{1} v_{2}+b_{2} = 0 \end{equation}

Setting each coefficients in (8E) to zero gives the following equations to solve

\begin{align*} b_{2}&=0\\ -a a_{1}&=0\\ -3 a b_{1}&=0\\ -3 a b_{2}&=0\\ -a^{2} a_{3}&=0\\ -2 b b_{1}&=0\\ -2 b b_{2}&=0\\ -2 a b a_{3}&=0\\ -b^{2} a_{3}-a a_{3}&=0\\ -2 a a_{2}-2 a b_{3}&=0\\ -b a_{2}-b b_{3}&=0 \end{align*}

Solving the above equations for the unknowns gives

\begin{align*} a_{1}&=0\\ a_{2}&=-b_{3}\\ a_{3}&=0\\ b_{1}&=0\\ b_{2}&=0\\ b_{3}&=b_{3} \end{align*}

Substituting the above solution in the anstaz (1E,2E) (using \(1\) as arbitrary value for any unknown in the RHS) gives

\begin{align*} \xi &= -x \\ \eta &= y \\ \end{align*}

The next step is to determine the canonical coordinates \(R,S\). The canonical coordinates map \(\left ( x,y\right ) \to \left ( R,S \right )\) where \(\left ( R,S \right )\) are the canonical coordinates which make the original ode become a quadrature and hence solved by integration.

The characteristic pde which is used to find the canonical coordinates is

\begin{align*} \frac {d x}{\xi } &= \frac {d y}{\eta } = dS \tag {1} \end{align*}

The above comes from the requirements that \(\left ( \xi \frac {\partial }{\partial x} + \eta \frac {\partial }{\partial y}\right ) S(x,y) = 1\). Starting with the first pair of ode’s in (1) gives an ode to solve for the independent variable \(R\) in the canonical coordinates, where \(S(R)\). Therefore

\begin{align*} \frac {dy}{dx} &= \frac {\eta }{\xi }\\ &= \frac {y}{-x}\\ &= -\frac {y}{x} \end{align*}

This is easily solved to give

\begin{align*} y = \frac {c_1}{x} \end{align*}

Where now the coordinate \(R\) is taken as the constant of integration. Hence

\begin{align*} R &= x y \end{align*}

And \(S\) is found from

\begin{align*} dS &= \frac {dx}{\xi } \\ &= \frac {dx}{-x} \end{align*}

Integrating gives

\begin{align*} S &= \int { \frac {dx}{T}}\\ &= -\ln \left (x \right ) \end{align*}

Where the constant of integration is set to zero as we just need one solution. Now that \(R,S\) are found, we need to setup the ode in these coordinates. This is done by evaluating

\begin{align*} \frac {dS}{dR} &= \frac { S_{x} + \omega (x,y) S_{y} }{ R_{x} + \omega (x,y) R_{y} }\tag {2} \end{align*}

Where in the above \(R_{x},R_{y},S_{x},S_{y}\) are all partial derivatives and \(\omega (x,y)\) is the right hand side of the original ode given by

\begin{align*} \omega (x,y) &= a x \,y^{3}+y^{2} b \end{align*}

Evaluating all the partial derivatives gives

\begin{align*} R_{x} &= y\\ R_{y} &= x\\ S_{x} &= -\frac {1}{x}\\ S_{y} &= 0 \end{align*}

Substituting all the above in (2) and simplifying gives the ode in canonical coordinates.

\begin{align*} \frac {dS}{dR} &= -\frac {1}{x y \left (x^{2} a \,y^{2}+b x y +1\right )}\tag {2A} \end{align*}

We now need to express the RHS as function of \(R\) only. This is done by solving for \(x,y\) in terms of \(R,S\) from the result obtained earlier and simplifying. This gives

\begin{align*} \frac {dS}{dR} &= -\frac {1}{R \left (a \,R^{2}+b R +1\right )} \end{align*}

The above is a quadrature ode. This is the whole point of Lie symmetry method. It converts an ode, no matter how complicated it is, to one that can be solved by integration when the ode is in the canonical coordiates \(R,S\).

Since the ode has the form \(\frac {d}{d R}S \left (R \right )=f(R)\), then we only need to integrate \(f(R)\).

\begin{align*} \int {dS} &= \int {-\frac {1}{R \left (a \,R^{2}+b R +1\right )}\, dR}\\ S \left (R \right ) &= \frac {\ln \left (a \,R^{2}+b R +1\right )}{2}-\frac {b \,\operatorname {arctanh}\left (\frac {2 R a +b}{\sqrt {b^{2}-4 a}}\right )}{\sqrt {b^{2}-4 a}}-\ln \left (R \right ) + c_2 \end{align*}

\begin{align*} S \left (R \right )&= \frac {\ln \left (a \,R^{2}+b R +1\right )}{2}-\frac {b \,\operatorname {arctanh}\left (\frac {2 R a +b}{\sqrt {b^{2}-4 a}}\right )}{\sqrt {b^{2}-4 a}}-\ln \left (R \right )+c_2 \end{align*}

To complete the solution, we just need to transform the above back to \(x,y\) coordinates. This results in

\begin{align*} -\ln \left (x \right ) = \frac {\ln \left (x^{2} a y^{2}+b x y+1\right )}{2}-\frac {b \,\operatorname {arctanh}\left (\frac {2 y a x +b}{\sqrt {b^{2}-4 a}}\right )}{\sqrt {b^{2}-4 a}}-\ln \left (x y\right )+c_2 \end{align*}

Summary of solutions found

\begin{align*} -\ln \left (x \right ) &= \frac {\ln \left (x^{2} a y^{2}+b x y+1\right )}{2}-\frac {b \,\operatorname {arctanh}\left (\frac {2 y a x +b}{\sqrt {b^{2}-4 a}}\right )}{\sqrt {b^{2}-4 a}}-\ln \left (x y\right )+c_2 \\ \end{align*}

2.26.7.7 ✓ Maple. Time used: 0.024 (sec). Leaf size: 102

ode:=diff(y(x),x) = a*x*y(x)^3+b*y(x)^2; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {{\mathrm e}^{\operatorname {RootOf}\left (-2 \sqrt {b^{2}-4 a}\, b \,\operatorname {arctanh}\left (\frac {2 a \,{\mathrm e}^{\textit {\_Z}}+b}{\sqrt {b^{2}-4 a}}\right )+\ln \left (x^{2} \left (a \,{\mathrm e}^{2 \textit {\_Z}}+b \,{\mathrm e}^{\textit {\_Z}}+1\right )\right ) b^{2}-2 c_1 \,b^{2}-2 \textit {\_Z} \,b^{2}-4 \ln \left (x^{2} \left (a \,{\mathrm e}^{2 \textit {\_Z}}+b \,{\mathrm e}^{\textit {\_Z}}+1\right )\right ) a +8 c_1 a +8 \textit {\_Z} a \right )}}{x} \]

Maple trace

Methods for first order ODEs: 
--- Trying classification methods --- 
trying a quadrature 
trying 1st order linear 
trying Bernoulli 
trying separable 
trying inverse linear 
trying homogeneous types: 
trying homogeneous G 
<- homogeneous successful
 

Maple step by step

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=a x y \left (x \right )^{3}+b y \left (x \right )^{2} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=a x y \left (x \right )^{3}+b y \left (x \right )^{2} \end {array} \]

2.26.7.8 ✓ Mathematica. Time used: 0.11 (sec). Leaf size: 94

ode=D[y[x],x]==a*x*y[x]^3+b*y[x]^2; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [\frac {b^2 \left (-\log \left (\frac {a x y(x) (a x y(x)+b)+a}{a^2 x^2 y(x)^2}\right )-\frac {2 \arctan \left (\frac {2 a x y(x)+b}{b \sqrt {\frac {4 a}{b^2}-1}}\right )}{\sqrt {\frac {4 a}{b^2}-1}}\right )}{2 a}=\frac {b^2 \log (x)}{a}+c_1,y(x)\right ] \]

2.26.7.9 ✓ Sympy. Time used: 5.705 (sec). Leaf size: 416

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
y = Function("y") 
ode = Eq(-a*x*y(x)**3 - b*y(x)**2 + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ C_{1} - \frac {\left (- \frac {b \sqrt {- 4 a + b^{2}}}{4 a - b^{2}} - 1\right ) \log {\left (x y{\left (x \right )} + \frac {6 a^{2} \left (- \frac {b \sqrt {- 4 a + b^{2}}}{4 a - b^{2}} - 1\right )^{2} - 6 a^{2} \left (- \frac {b \sqrt {- 4 a + b^{2}}}{4 a - b^{2}} - 1\right ) - 12 a^{2} - \frac {7 a b^{2} \left (- \frac {b \sqrt {- 4 a + b^{2}}}{4 a - b^{2}} - 1\right )^{2}}{2} + \frac {3 a b^{2} \left (- \frac {b \sqrt {- 4 a + b^{2}}}{4 a - b^{2}} - 1\right )}{2} + 11 a b^{2} + \frac {b^{4} \left (- \frac {b \sqrt {- 4 a + b^{2}}}{4 a - b^{2}} - 1\right )^{2}}{2} - 2 b^{4}}{a b \left (9 a - 2 b^{2}\right )} \right )}}{2} - \frac {\left (\frac {b \sqrt {- 4 a + b^{2}}}{4 a - b^{2}} - 1\right ) \log {\left (x y{\left (x \right )} + \frac {6 a^{2} \left (\frac {b \sqrt {- 4 a + b^{2}}}{4 a - b^{2}} - 1\right )^{2} - 6 a^{2} \left (\frac {b \sqrt {- 4 a + b^{2}}}{4 a - b^{2}} - 1\right ) - 12 a^{2} - \frac {7 a b^{2} \left (\frac {b \sqrt {- 4 a + b^{2}}}{4 a - b^{2}} - 1\right )^{2}}{2} + \frac {3 a b^{2} \left (\frac {b \sqrt {- 4 a + b^{2}}}{4 a - b^{2}} - 1\right )}{2} + 11 a b^{2} + \frac {b^{4} \left (\frac {b \sqrt {- 4 a + b^{2}}}{4 a - b^{2}} - 1\right )^{2}}{2} - 2 b^{4}}{a b \left (9 a - 2 b^{2}\right )} \right )}}{2} + \log {\left (x \right )} - \log {\left (x y{\left (x \right )} \right )} = 0 \]