2.2.8 Problem (i)

Internal problem ID [20974]
Book : Ordinary Differential Equations. By Wolfgang Walter. Graduate texts in Mathematics. Springer. NY. QA372.W224 1998
Section : Chapter 1. First order equations: Some integrable cases. Excercises XIII at page 24
Problem number : (i)
Date solved : Saturday, November 29, 2025 at 01:18:17 AM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

Solved using first_order_ode_isobaric

Time used: 44.257 (sec)

Solve

\begin{align*} y&=y^{\prime } x -\sqrt {x^{2}+y^{2}} \\ \end{align*}

Solving for \(y'\) gives

\begin{align*} \tag{1} y' &= \frac {y+\sqrt {x^{2}+y^{2}}}{x} \\ \end{align*}

Each of the above ode’s is now solved 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

\[ f(x,y) = \frac {y+\sqrt {x^{2}+y^{2}}}{x}\tag {2} \]

\(m\) 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 \\ &=u x \end{align*}

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

\[ u \left (x \right )+x u^{\prime }\left (x \right ) = \frac {x u \left (x \right )+\sqrt {x^{2}+x^{2} u \left (x \right )^{2}}}{x} \]

The ode

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

is separable as it can be written as

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

Where

\begin{align*} f(x) &= \frac {1}{x}\\ g(u) &= \sqrt {u^{2}+1} \end{align*}

Integrating gives

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

\[ \operatorname {arcsinh}\left (u \left (x \right )\right )=\ln \left (x \right )+c_1 \]

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

\[ \sqrt {u^{2}+1}=0 \]

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

\begin{align*} u \left (x \right )&=-i\\ u \left (x \right )&=i \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*} \operatorname {arcsinh}\left (u \left (x \right )\right ) &= \ln \left (x \right )+c_1 \\ u \left (x \right ) &= -i \\ u \left (x \right ) &= i \\ \end{align*}

Converting \(\operatorname {arcsinh}\left (u \left (x \right )\right ) = \ln \left (x \right )+c_1\) back to \(y\) gives

\begin{align*} \operatorname {arcsinh}\left (\frac {y}{x}\right ) = \ln \left (x \right )+c_1 \end{align*}

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

\begin{align*} \frac {y}{x} = -i \end{align*}

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

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

Solving for \(y\) gives

\begin{align*} \operatorname {arcsinh}\left (\frac {y}{x}\right ) &= \ln \left (x \right )+c_1 \\ y &= -i x \\ y &= i x \\ \end{align*}

The solution

\[ \operatorname {arcsinh}\left (\frac {y}{x}\right ) = \ln \left (x \right )+c_1 \]

was found not to satisfy the ode or the IC. Hence it is removed.

Figure 2.41: Slope field \(y = y^{\prime } x -\sqrt {x^{2}+y^{2}}\)

Summary of solutions found

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

Solved using first_order_ode_homog_A

Time used: 49.945 (sec)

Solve

\begin{align*} y&=y^{\prime } x -\sqrt {x^{2}+y^{2}} \\ \end{align*}

In canonical form, the ODE is

\begin{align*} y' &= F(x,y)\\ &= \frac {y +\sqrt {x^{2}+y^{2}}}{x}\tag {1} \end{align*}

An ode of the form \(y' = \frac {M(x,y)}{N(x,y)}\) is called homogeneous if the functions \(M(x,y)\) and \(N(x,y)\) are both homogeneous functions and of the same order. Recall that a function \(f(x,y)\) is homogeneous of order \(n\) if

\[ f(t^n x, t^n y)= t^n f(x,y) \]

In this case, it can be seen that both \(M=y +\sqrt {x^{2}+y^{2}}\) and \(N=x\) are both homogeneous and of the same order \(n=1\). Therefore this is a homogeneous ode. Since this ode is homogeneous, it is converted to separable ODE using the substitution \(u=\frac {y}{x}\), or \(y=ux\). Hence

\[ \frac { \mathop {\mathrm {d}y}}{\mathop {\mathrm {d}x}}= \frac { \mathop {\mathrm {d}u}}{\mathop {\mathrm {d}x}}x + u \]

Applying the transformation \(y=ux\) to the above ODE in (1) gives

\begin{align*} \frac { \mathop {\mathrm {d}u}}{\mathop {\mathrm {d}x}}x + u &= u +\sqrt {u^{2}+1}\\ \frac { \mathop {\mathrm {d}u}}{\mathop {\mathrm {d}x}} &= \frac {\sqrt {u \left (x \right )^{2}+1}}{x} \end{align*}

Or

\[ u^{\prime }\left (x \right )-\frac {\sqrt {u \left (x \right )^{2}+1}}{x} = 0 \]

Or

\[ u^{\prime }\left (x \right ) x -\sqrt {u \left (x \right )^{2}+1} = 0 \]

Which is now solved as separable in \(u \left (x \right )\).

The ode

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

is separable as it can be written as

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

Where

\begin{align*} f(x) &= \frac {1}{x}\\ g(u) &= \sqrt {u^{2}+1} \end{align*}

Integrating gives

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

\[ \operatorname {arcsinh}\left (u \left (x \right )\right )=\ln \left (x \right )+c_1 \]

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

\[ \sqrt {u^{2}+1}=0 \]

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

\begin{align*} u \left (x \right )&=-i\\ u \left (x \right )&=i \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*} \operatorname {arcsinh}\left (u \left (x \right )\right ) &= \ln \left (x \right )+c_1 \\ u \left (x \right ) &= -i \\ u \left (x \right ) &= i \\ \end{align*}

Converting \(\operatorname {arcsinh}\left (u \left (x \right )\right ) = \ln \left (x \right )+c_1\) back to \(y\) gives

\begin{align*} \operatorname {arcsinh}\left (\frac {y}{x}\right ) = \ln \left (x \right )+c_1 \end{align*}

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

\begin{align*} y = -i x \end{align*}

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

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

The solution

\[ \operatorname {arcsinh}\left (\frac {y}{x}\right ) = \ln \left (x \right )+c_1 \]

was found not to satisfy the ode or the IC. Hence it is removed.

Figure 2.42: Slope field \(y = y^{\prime } x -\sqrt {x^{2}+y^{2}}\)

Summary of solutions found

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

Solved using first_order_ode_dAlembert

Time used: 0.498 (sec)

Solve

\begin{align*} y&=y^{\prime } x -\sqrt {x^{2}+y^{2}} \\ \end{align*}

Let \(p=y^{\prime }\) the ode becomes

\begin{align*} y = p x -\sqrt {x^{2}+y^{2}} \end{align*}

Solving for \(y\) from the above results in

\begin{align*} \tag{1} y &= \frac {x \left (p^{2}-1\right )}{2 p} \\ \end{align*}

This has the form

\begin{align*} y=x f(p)+g(p)\tag {*} \end{align*}

Where \(f,g\) are functions of \(p=y'(x)\). The above ode is dAlembert ode which is now solved.

Taking derivative of (*) w.r.t. \(x\) gives

\begin{align*} p &= f+(x f'+g') \frac {dp}{dx}\\ p-f &= (x f'+g') \frac {dp}{dx}\tag {2} \end{align*}

Comparing the form \(y=x f + g\) to (1A) shows that

\begin{align*} f &= \frac {p^{2}-1}{2 p}\\ g &= 0 \end{align*}

Hence (2) becomes

\begin{equation} \tag{2A} p -\frac {p^{2}-1}{2 p} = \left (\frac {x}{2}+\frac {x}{2 p^{2}}\right ) p^{\prime }\left (x \right ) \end{equation}

The singular solution is found by setting \(\frac {dp}{dx}=0\) in the above which gives

\begin{align*} p -\frac {p^{2}-1}{2 p} = 0 \end{align*}

Solving the above for \(p\) results in

\begin{align*} p_{1} &=i\\ p_{2} &=-i \end{align*}

Substituting these in (1A) and keeping singular solution that verifies the ode gives

\begin{align*} y = i x\\ y = -i x \end{align*}

The general solution is found when \( \frac { \mathop {\mathrm {d}p}}{\mathop {\mathrm {d}x}}\neq 0\). From eq. (2A). This results in

\begin{equation} \tag{3} p^{\prime }\left (x \right ) = \frac {p \left (x \right )-\frac {p \left (x \right )^{2}-1}{2 p \left (x \right )}}{\frac {x}{2}+\frac {x}{2 p \left (x \right )^{2}}} \end{equation}

This ODE is now solved for \(p \left (x \right )\). No inversion is needed.

The ode

\begin{equation} p^{\prime }\left (x \right ) = \frac {p \left (x \right )}{x} \end{equation}

is separable as it can be written as

\begin{align*} p^{\prime }\left (x \right )&= \frac {p \left (x \right )}{x}\\ &= f(x) g(p) \end{align*}

Where

\begin{align*} f(x) &= \frac {1}{x}\\ g(p) &= p \end{align*}

Integrating gives

\begin{align*} \int { \frac {1}{g(p)} \,dp} &= \int { f(x) \,dx} \\ \int { \frac {1}{p}\,dp} &= \int { \frac {1}{x} \,dx} \\ \end{align*}

\[ \ln \left (p \left (x \right )\right )=\ln \left (x \right )+c_1 \]

Taking the exponential of both sides the solution becomes

\[ p \left (x \right ) = c_1 x \]

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

\[ p=0 \]

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

\begin{align*} p \left (x \right )&=0 \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*} p \left (x \right ) &= c_1 x \\ p \left (x \right ) &= 0 \\ \end{align*}

Substituing the above solution for \(p\) in (2A) gives

\begin{align*} y &= \frac {c_1^{2} x^{2}-1}{2 c_1} \\ \end{align*}

Figure 2.43: Slope field \(y = y^{\prime } x -\sqrt {x^{2}+y^{2}}\)

Summary of solutions found

\begin{align*} y &= -i x \\ y &= i x \\ y &= \frac {c_1^{2} x^{2}-1}{2 c_1} \\ \end{align*}

Solved using first_order_ode_homog_type_maple_C

Time used: 1.770 (sec)

Solve

\begin{align*} y&=y^{\prime } x -\sqrt {x^{2}+y^{2}} \\ \end{align*}

Let \(Y = y -y_{0}\) and \(X = x -x_{0}\) then the above is transformed to new ode in \(Y(X)\)

\[ \frac {d}{d X}Y \left (X \right ) = \frac {Y \left (X \right )+y_{0} +\sqrt {\left (X +x_{0} \right )^{2}+\left (Y \left (X \right )+y_{0} \right )^{2}}}{X +x_{0}} \]

Solving for possible values of \(x_{0}\) and \(y_{0}\) which makes the above ode a homogeneous ode results in

\begin{align*} x_{0}&=0\\ y_{0}&=0 \end{align*}

Using these values now it is possible to easily solve for \(Y \left (X \right )\). The above ode now becomes

\begin{align*} \frac {d}{d X}Y \left (X \right ) = \frac {Y \left (X \right )+\sqrt {X^{2}+Y \left (X \right )^{2}}}{X} \end{align*}

In canonical form, the ODE is

\begin{align*} Y' &= F(X,Y)\\ &= \frac {Y +\sqrt {X^{2}+Y^{2}}}{X}\tag {1} \end{align*}

An ode of the form \(Y' = \frac {M(X,Y)}{N(X,Y)}\) is called homogeneous if the functions \(M(X,Y)\) and \(N(X,Y)\) are both homogeneous functions and of the same order. Recall that a function \(f(X,Y)\) is homogeneous of order \(n\) if

\[ f(t^n X, t^n Y)= t^n f(X,Y) \]

In this case, it can be seen that both \(M=Y +\sqrt {X^{2}+Y^{2}}\) and \(N=X\) are both homogeneous and of the same order \(n=1\). Therefore this is a homogeneous ode. Since this ode is homogeneous, it is converted to separable ODE using the substitution \(u=\frac {Y}{X}\), or \(Y=uX\). Hence

\[ \frac { \mathop {\mathrm {d}Y}}{\mathop {\mathrm {d}X}}= \frac { \mathop {\mathrm {d}u}}{\mathop {\mathrm {d}X}}X + u \]

Applying the transformation \(Y=uX\) to the above ODE in (1) gives

\begin{align*} \frac { \mathop {\mathrm {d}u}}{\mathop {\mathrm {d}X}}X + u &= u +\sqrt {u^{2}+1}\\ \frac { \mathop {\mathrm {d}u}}{\mathop {\mathrm {d}X}} &= \frac {\sqrt {u \left (X \right )^{2}+1}}{X} \end{align*}

Or

\[ \frac {d}{d X}u \left (X \right )-\frac {\sqrt {u \left (X \right )^{2}+1}}{X} = 0 \]

Or

\[ \left (\frac {d}{d X}u \left (X \right )\right ) X -\sqrt {u \left (X \right )^{2}+1} = 0 \]

Which is now solved as separable in \(u \left (X \right )\).

The ode

\begin{equation} \frac {d}{d X}u \left (X \right ) = \frac {\sqrt {u \left (X \right )^{2}+1}}{X} \end{equation}

is separable as it can be written as

\begin{align*} \frac {d}{d X}u \left (X \right )&= \frac {\sqrt {u \left (X \right )^{2}+1}}{X}\\ &= f(X) g(u) \end{align*}

Where

\begin{align*} f(X) &= \frac {1}{X}\\ g(u) &= \sqrt {u^{2}+1} \end{align*}

Integrating gives

\begin{align*} \int { \frac {1}{g(u)} \,du} &= \int { f(X) \,dX} \\ \int { \frac {1}{\sqrt {u^{2}+1}}\,du} &= \int { \frac {1}{X} \,dX} \\ \end{align*}

\[ \operatorname {arcsinh}\left (u \left (X \right )\right )=\ln \left (X \right )+c_1 \]

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

\[ \sqrt {u^{2}+1}=0 \]

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

\begin{align*} u \left (X \right )&=-i\\ u \left (X \right )&=i \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*} \operatorname {arcsinh}\left (u \left (X \right )\right ) &= \ln \left (X \right )+c_1 \\ u \left (X \right ) &= -i \\ u \left (X \right ) &= i \\ \end{align*}

Converting \(\operatorname {arcsinh}\left (u \left (X \right )\right ) = \ln \left (X \right )+c_1\) back to \(Y \left (X \right )\) gives

\begin{align*} \operatorname {arcsinh}\left (\frac {Y \left (X \right )}{X}\right ) = \ln \left (X \right )+c_1 \end{align*}

Converting \(u \left (X \right ) = -i\) back to \(Y \left (X \right )\) gives

\begin{align*} Y \left (X \right ) = -i X \end{align*}

Converting \(u \left (X \right ) = i\) back to \(Y \left (X \right )\) gives

\begin{align*} Y \left (X \right ) = i X \end{align*}

Using the solution for \(Y(X)\)

\begin{align*} \operatorname {arcsinh}\left (\frac {Y \left (X \right )}{X}\right ) = \ln \left (X \right )+c_1\tag {A} \end{align*}

And replacing back terms in the above solution using

\begin{align*} Y &= y +y_{0}\\ X &= x_{0} +x \end{align*}

Or

\begin{align*} Y &= y\\ X &= x \end{align*}

Then the solution in \(y\) becomes using EQ (A)

\begin{align*} \operatorname {arcsinh}\left (\frac {y}{x}\right ) = \ln \left (x \right )+c_1 \end{align*}

Using the solution for \(Y(X)\)

\begin{align*} Y \left (X \right ) = -i X\tag {A} \end{align*}

And replacing back terms in the above solution using

\begin{align*} Y &= y +y_{0}\\ X &= x_{0} +x \end{align*}

Or

\begin{align*} Y &= y\\ X &= x \end{align*}

Then the solution in \(y\) becomes using EQ (A)

\begin{align*} y = -i x \end{align*}

Using the solution for \(Y(X)\)

\begin{align*} Y \left (X \right ) = i X\tag {A} \end{align*}

And replacing back terms in the above solution using

\begin{align*} Y &= y +y_{0}\\ X &= x_{0} +x \end{align*}

Or

\begin{align*} Y &= y\\ X &= x \end{align*}

Then the solution in \(y\) becomes using EQ (A)

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

The solution

\[ \operatorname {arcsinh}\left (\frac {y}{x}\right ) = \ln \left (x \right )+c_1 \]

was found not to satisfy the ode or the IC. Hence it is removed.

Figure 2.44: Slope field \(y = y^{\prime } x -\sqrt {x^{2}+y^{2}}\)

Summary of solutions found

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

Solved using first_order_ode_LIE

Time used: 3.779 (sec)

Solve

\begin{align*} y&=y^{\prime } x -\sqrt {x^{2}+y^{2}} \\ \end{align*}

Writing the ode as

\begin{align*} y^{\prime }&=\frac {y +\sqrt {x^{2}+y^{2}}}{x}\\ 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. Using these anstaz

\begin{align*} \tag{1E} \xi &= \frac {A x +B y}{C x} \\ \tag{2E} \eta &= 1 \\ \end{align*}

Where the unknown coefficients are

\[ \{A, B, C\} \]

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

\begin{equation} \tag{5E} \frac {\left (y +\sqrt {x^{2}+y^{2}}\right ) \left (-\frac {A}{C x}+\frac {A x +B y}{C \,x^{2}}\right )}{x}-\frac {\left (y +\sqrt {x^{2}+y^{2}}\right )^{2} B}{x^{3} C}-\frac {\left (\frac {1}{\sqrt {x^{2}+y^{2}}}-\frac {y +\sqrt {x^{2}+y^{2}}}{x^{2}}\right ) \left (A x +B y \right )}{C x}-\frac {1+\frac {y}{\sqrt {x^{2}+y^{2}}}}{x} = 0 \end{equation}

Putting the above in normal form gives

\[ \frac {A \sqrt {x^{2}+y^{2}}\, x y +A x \,y^{2}-B \left (x^{2}+y^{2}\right )^{{3}/{2}}+B \sqrt {x^{2}+y^{2}}\, y^{2}-B \,x^{2} y -x^{2} \sqrt {x^{2}+y^{2}}\, C -C \,x^{2} y}{\sqrt {x^{2}+y^{2}}\, x^{3} C} = 0 \]

Setting the numerator to zero gives

\begin{equation} \tag{6E} A \sqrt {x^{2}+y^{2}}\, x y +A x \,y^{2}-B \left (x^{2}+y^{2}\right )^{{3}/{2}}+B \sqrt {x^{2}+y^{2}}\, y^{2}-B \,x^{2} y -x^{2} \sqrt {x^{2}+y^{2}}\, C -C \,x^{2} y = 0 \end{equation}

Simplifying the above gives

\begin{equation} \tag{6E} A \left (x^{2}+y^{2}\right ) x +A \sqrt {x^{2}+y^{2}}\, x y -A \,x^{3}-B \left (x^{2}+y^{2}\right )^{{3}/{2}}+B \sqrt {x^{2}+y^{2}}\, y^{2}-B \,x^{2} y -x^{2} \sqrt {x^{2}+y^{2}}\, C -C \,x^{2} y = 0 \end{equation}

Since the PDE has radicals, simplifying gives

\[ x \left (A \sqrt {x^{2}+y^{2}}\, y +A \,y^{2}-B \sqrt {x^{2}+y^{2}}\, x -B x y -C \sqrt {x^{2}+y^{2}}\, x -x C y \right ) = 0 \]

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

\[ \left \{x, y, \sqrt {x^{2}+y^{2}}\right \} \]

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

\[ \left \{x = v_{1}, y = v_{2}, \sqrt {x^{2}+y^{2}} = v_{3}\right \} \]

The above PDE (6E) now becomes

\begin{equation} \tag{7E} v_{1} \left (A v_{2}^{2}+A v_{3} v_{2}-B v_{1} v_{2}-B v_{3} v_{1}-v_{1} C v_{2}-C v_{3} v_{1}\right ) = 0 \end{equation}

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

\[ \{v_{1}, v_{2}, v_{3}\} \]

Equation (7E) now becomes

\begin{equation} \tag{8E} \left (-B -C \right ) v_{2} v_{1}^{2}+\left (-B -C \right ) v_{3} v_{1}^{2}+A v_{2}^{2} v_{1}+A v_{2} v_{3} v_{1} = 0 \end{equation}

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

\begin{align*} A&=0\\ -B -C&=0 \end{align*}

Solving the above equations for the unknowns gives

\begin{align*} A&=0\\ B&=-C\\ C&=C \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 &= -\frac {y}{x} \\ \eta &= 1 \\ \end{align*}

Shifting is now applied to make \(\xi =0\) in order to simplify the rest of the computation

\begin{align*} \eta &= \eta - \omega \left (x,y\right ) \xi \\ &= 1 - \left (\frac {y +\sqrt {x^{2}+y^{2}}}{x}\right ) \left (-\frac {y}{x}\right ) \\ &= \frac {\sqrt {x^{2}+y^{2}}\, y +x^{2}+y^{2}}{x^{2}}\\ \xi &= 0 \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 \(S(R)\). Since \(\xi =0\) then in this special case

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

\(S\) is found from

\begin{align*} S &= \int { \frac {1}{\eta }} dy\\ &= \int { \frac {1}{\frac {\sqrt {x^{2}+y^{2}}\, y +x^{2}+y^{2}}{x^{2}}}} dy \end{align*}

Which results in

\begin{align*} S&= x^{2} \left (\frac {y}{x^{2}}-\frac {\sqrt {x^{2}+y^{2}}}{x^{2}}\right ) \end{align*}

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) &= \frac {y +\sqrt {x^{2}+y^{2}}}{x} \end{align*}

Evaluating all the partial derivatives gives

\begin{align*} R_{x} &= 1\\ R_{y} &= 0\\ S_{x} &= -\frac {x}{\sqrt {x^{2}+y^{2}}}\\ S_{y} &= 1-\frac {y}{\sqrt {x^{2}+y^{2}}} \end{align*}

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

\begin{align*} \frac {dS}{dR} &= 0\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} &= 0 \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 {0\, dR} + c_2 \\ S \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*} y-\sqrt {x^{2}+y^{2}} = c_2 \end{align*}

The following diagram shows solution curves of the original ode and how they transform in the canonical coordinates space using the mapping shown.

Original ode in \(x,y\) coordinates	Canonical coordinates transformation	ODE in canonical coordinates \((R,S)\)
\( \frac {dy}{dx} = \frac {y +\sqrt {x^{2}+y^{2}}}{x}\)		\( \frac {d S}{d R} = 0\)
	\(\!\begin {aligned} R&= x\\ S&= -\sqrt {x^{2}+y^{2}}+y \end {aligned} \)

Solving for \(y\) gives

\begin{align*} y &= \frac {c_2^{2}-x^{2}}{2 c_2} \\ \end{align*}

Figure 2.45: Slope field \(y = y^{\prime } x -\sqrt {x^{2}+y^{2}}\)

Summary of solutions found

\begin{align*} y &= \frac {c_2^{2}-x^{2}}{2 c_2} \\ \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 26

ode:=y(x) = diff(y(x),x)*x-(x^2+y(x)^2)^(1/2); 
dsolve(ode,y(x), singsol=all);
 

\[ \frac {-c_1 \,x^{2}+y+\sqrt {x^{2}+y^{2}}}{x^{2}} = 0 \]

Maple trace

Methods for first order ODEs: 
--- Trying classification methods --- 
trying homogeneous types: 
trying homogeneous G 
1st order, trying the canonical coordinates of the invariance group 
<- 1st order, canonical coordinates successful 
<- homogeneous successful
 

Maple step by step

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

✓ Mathematica. Time used: 0.18 (sec). Leaf size: 13

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

\begin{align*} y(x)&\to x \sinh (\log (x)+c_1) \end{align*}

✓ Sympy. Time used: 0.733 (sec). Leaf size: 12

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

\[ y{\left (x \right )} = - x \sinh {\left (C_{1} - \log {\left (x \right )} \right )} \]