2.1.2 Internal
problem
ID
[20963]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
XII
at
page
23 Problem
number
:
(b)Date
solved
:
Saturday, November 29, 2025 at 01:10:19 AMCAS
classification
:
[[_homogeneous, `class C`], _dAlembert]
Time used: 0.871 (sec)
Solve
\begin{align*}
y^{\prime }&=\frac {y+1}{x +2}+{\mathrm e}^{\frac {y+1}{x +2}} \\
\end{align*}
Let
\(Y = y -y_{0}\) and
\(X = x -x_{0}\) then the above is transformed to new ode in
\(Y(X)\) \[
\frac {d}{d X}Y \left (X \right ) = \frac {{\mathrm e}^{\frac {Y \left (X \right )+y_{0} +1}{X +x_{0} +2}} \left (X +x_{0} \right )+2 \,{\mathrm e}^{\frac {Y \left (X \right )+y_{0} +1}{X +x_{0} +2}}+Y \left (X \right )+y_{0} +1}{X +x_{0} +2}
\]
Solving for possible values of
\(x_{0}\) and
\(y_{0}\)
which makes the above ode a homogeneous ode results in
\begin{align*} x_{0}&=-2\\ y_{0}&=-1 \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 {{\mathrm e}^{\frac {Y \left (X \right )}{X}} {\mathrm e}^{-\frac {1}{X}} {\mathrm e}^{\frac {1}{X}} X +Y \left (X \right )}{X} \end{align*}
In canonical form, the ODE is
\begin{align*} Y' &= F(X,Y)\\ &= \frac {{\mathrm e}^{\frac {Y}{X}} {\mathrm e}^{-\frac {1}{X}} {\mathrm e}^{\frac {1}{X}} X +Y}{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
\(M={\mathrm e}^{\frac {Y}{X}} {\mathrm e}^{-\frac {1}{X}} {\mathrm e}^{\frac {1}{X}} X +Y\) 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 &= {\mathrm e}^{u}+u\\ \frac { \mathop {\mathrm {d}u}}{\mathop {\mathrm {d}X}} &= \frac {{\mathrm e}^{u \left (X \right )}}{X} \end{align*}
Or
\[ \frac {d}{d X}u \left (X \right )-\frac {{\mathrm e}^{u \left (X \right )}}{X} = 0 \]
Or
\[ \left (\frac {d}{d X}u \left (X \right )\right ) X -{\mathrm e}^{u \left (X \right )} = 0 \]
Which is now solved as separable in
\(u \left (X \right )\) .
The ode
\begin{equation}
\frac {d}{d X}u \left (X \right ) = \frac {{\mathrm e}^{u \left (X \right )}}{X}
\end{equation}
is separable as it can be written as
\begin{align*} \frac {d}{d X}u \left (X \right )&= \frac {{\mathrm e}^{u \left (X \right )}}{X}\\ &= f(X) g(u) \end{align*}
Where
\begin{align*} f(X) &= \frac {1}{X}\\ g(u) &= {\mathrm e}^{u} \end{align*}
Integrating gives
\begin{align*}
\int { \frac {1}{g(u)} \,du} &= \int { f(X) \,dX} \\
\int { {\mathrm e}^{-u}\,du} &= \int { \frac {1}{X} \,dX} \\
\end{align*}
\[
-{\mathrm e}^{-u \left (X \right )}=\ln \left (X \right )+c_1
\]
Converting
\(-{\mathrm e}^{-u \left (X \right )} = \ln \left (X \right )+c_1\) back to
\(Y \left (X \right )\) gives
\begin{align*} -{\mathrm e}^{-\frac {Y \left (X \right )}{X}} = \ln \left (X \right )+c_1 \end{align*}
Using the solution for \(Y(X)\)
\begin{align*} -{\mathrm e}^{-\frac {Y \left (X \right )}{X}} = \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 -1\\ X &= x -2 \end{align*}
Then the solution in \(y\) becomes using EQ (A)
\begin{align*} -{\mathrm e}^{-\frac {y+1}{x +2}} = \ln \left (x +2\right )+c_1 \end{align*}
Simplifying the above gives
\begin{align*}
-{\mathrm e}^{\frac {-y-1}{x +2}} &= \ln \left (x +2\right )+c_1 \\
\end{align*}
Solving for
\(y\) gives
\begin{align*}
y &= -\ln \left (-\ln \left (x +2\right )-c_1 \right ) x -2 \ln \left (-\ln \left (x +2\right )-c_1 \right )-1 \\
\end{align*}
Figure 2.3: Slope field \(y^{\prime } = \frac {y+1}{x +2}+{\mathrm e}^{\frac {y+1}{x +2}}\) Summary of solutions found
\begin{align*}
y &= -\ln \left (-\ln \left (x +2\right )-c_1 \right ) x -2 \ln \left (-\ln \left (x +2\right )-c_1 \right )-1 \\
\end{align*}
Time used: 1.483 (sec)
Solve
\begin{align*}
y^{\prime }&=\frac {y+1}{x +2}+{\mathrm e}^{\frac {y+1}{x +2}} \\
\end{align*}
Writing the ode as
\begin{align*} y^{\prime }&=\frac {{\mathrm e}^{\frac {y +1}{x +2}} x +2 \,{\mathrm e}^{\frac {y +1}{x +2}}+y +1}{x +2}\\ 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}+\frac {\left ({\mathrm e}^{\frac {y +1}{x +2}} x +2 \,{\mathrm e}^{\frac {y +1}{x +2}}+y +1\right ) \left (b_{3}-a_{2}\right )}{x +2}-\frac {\left ({\mathrm e}^{\frac {y +1}{x +2}} x +2 \,{\mathrm e}^{\frac {y +1}{x +2}}+y +1\right )^{2} a_{3}}{\left (x +2\right )^{2}}-\left (\frac {-\frac {\left (y +1\right ) {\mathrm e}^{\frac {y +1}{x +2}} x}{\left (x +2\right )^{2}}+{\mathrm e}^{\frac {y +1}{x +2}}-\frac {2 \left (y +1\right ) {\mathrm e}^{\frac {y +1}{x +2}}}{\left (x +2\right )^{2}}}{x +2}-\frac {{\mathrm e}^{\frac {y +1}{x +2}} x +2 \,{\mathrm e}^{\frac {y +1}{x +2}}+y +1}{\left (x +2\right )^{2}}\right ) \left (x a_{2}+y a_{3}+a_{1}\right )-\frac {\left (\frac {{\mathrm e}^{\frac {y +1}{x +2}} x}{x +2}+\frac {2 \,{\mathrm e}^{\frac {y +1}{x +2}}}{x +2}+1\right ) \left (x b_{2}+y b_{3}+b_{1}\right )}{x +2} = 0
\end{equation}
Putting the above in normal form gives
\[
-\frac {-{\mathrm e}^{\frac {y +1}{x +2}} x y a_{2}+2 \,{\mathrm e}^{\frac {y +1}{x +2}} x y a_{3}+{\mathrm e}^{\frac {y +1}{x +2}} x y b_{3}-a_{1}+2 a_{2}+a_{3}+2 b_{1}-4 b_{2}-2 b_{3}+{\mathrm e}^{\frac {2 y +2}{x +2}} x^{2} a_{3}+4 \,{\mathrm e}^{\frac {2 y +2}{x +2}} x a_{3}+{\mathrm e}^{\frac {y +1}{x +2}} x^{2} a_{2}+{\mathrm e}^{\frac {y +1}{x +2}} x^{2} b_{2}-{\mathrm e}^{\frac {y +1}{x +2}} x^{2} b_{3}-{\mathrm e}^{\frac {y +1}{x +2}} y^{2} a_{3}+3 \,{\mathrm e}^{\frac {y +1}{x +2}} x a_{2}+2 \,{\mathrm e}^{\frac {y +1}{x +2}} x a_{3}+{\mathrm e}^{\frac {y +1}{x +2}} x b_{1}+2 \,{\mathrm e}^{\frac {y +1}{x +2}} x b_{2}-4 \,{\mathrm e}^{\frac {y +1}{x +2}} x b_{3}-{\mathrm e}^{\frac {y +1}{x +2}} y a_{1}+3 \,{\mathrm e}^{\frac {y +1}{x +2}} y a_{3}+2 \,{\mathrm e}^{\frac {y +1}{x +2}} y b_{3}+y a_{3}-2 x b_{2}+4 \,{\mathrm e}^{\frac {2 y +2}{x +2}} a_{3}+x b_{1}-x b_{3}-y a_{1}+2 y a_{2}-{\mathrm e}^{\frac {y +1}{x +2}} a_{1}+4 \,{\mathrm e}^{\frac {y +1}{x +2}} a_{2}+4 \,{\mathrm e}^{\frac {y +1}{x +2}} a_{3}+2 \,{\mathrm e}^{\frac {y +1}{x +2}} b_{1}-4 \,{\mathrm e}^{\frac {y +1}{x +2}} b_{3}}{\left (x +2\right )^{2}} = 0
\]
Setting the numerator to zero gives
\begin{equation}
\tag{6E} {\mathrm e}^{\frac {y +1}{x +2}} x y a_{2}-2 \,{\mathrm e}^{\frac {y +1}{x +2}} x y a_{3}-{\mathrm e}^{\frac {y +1}{x +2}} x y b_{3}+a_{1}-2 a_{2}-a_{3}-2 b_{1}+4 b_{2}+2 b_{3}-{\mathrm e}^{\frac {2 y +2}{x +2}} x^{2} a_{3}-4 \,{\mathrm e}^{\frac {2 y +2}{x +2}} x a_{3}-{\mathrm e}^{\frac {y +1}{x +2}} x^{2} a_{2}-{\mathrm e}^{\frac {y +1}{x +2}} x^{2} b_{2}+{\mathrm e}^{\frac {y +1}{x +2}} x^{2} b_{3}+{\mathrm e}^{\frac {y +1}{x +2}} y^{2} a_{3}-3 \,{\mathrm e}^{\frac {y +1}{x +2}} x a_{2}-2 \,{\mathrm e}^{\frac {y +1}{x +2}} x a_{3}-{\mathrm e}^{\frac {y +1}{x +2}} x b_{1}-2 \,{\mathrm e}^{\frac {y +1}{x +2}} x b_{2}+4 \,{\mathrm e}^{\frac {y +1}{x +2}} x b_{3}+{\mathrm e}^{\frac {y +1}{x +2}} y a_{1}-3 \,{\mathrm e}^{\frac {y +1}{x +2}} y a_{3}-2 \,{\mathrm e}^{\frac {y +1}{x +2}} y b_{3}-y a_{3}+2 x b_{2}-4 \,{\mathrm e}^{\frac {2 y +2}{x +2}} a_{3}-x b_{1}+x b_{3}+y a_{1}-2 y a_{2}+{\mathrm e}^{\frac {y +1}{x +2}} a_{1}-4 \,{\mathrm e}^{\frac {y +1}{x +2}} a_{2}-4 \,{\mathrm e}^{\frac {y +1}{x +2}} a_{3}-2 \,{\mathrm e}^{\frac {y +1}{x +2}} b_{1}+4 \,{\mathrm e}^{\frac {y +1}{x +2}} b_{3} = 0
\end{equation}
Simplifying the
above gives
\begin{equation}
\tag{6E} 2 \,{\mathrm e}^{\frac {y +1}{x +2}} x y a_{2}-7 \,{\mathrm e}^{\frac {y +1}{x +2}} x y a_{3}-4 \,{\mathrm e}^{\frac {y +1}{x +2}} x y b_{3}+{\mathrm e}^{\frac {y +1}{x +2}} x^{2} y a_{2}-2 \,{\mathrm e}^{\frac {y +1}{x +2}} x^{2} y a_{3}-{\mathrm e}^{\frac {y +1}{x +2}} x^{2} y b_{3}+{\mathrm e}^{\frac {y +1}{x +2}} x \,y^{2} a_{3}+{\mathrm e}^{\frac {y +1}{x +2}} x y a_{1}-2 \,{\mathrm e}^{\frac {y +1}{x +2}} x^{2} a_{3}-{\mathrm e}^{\frac {y +1}{x +2}} x^{2} b_{1}+{\mathrm e}^{\frac {y +1}{x +2}} x a_{1}-{\mathrm e}^{\frac {y +1}{x +2}} x^{3} a_{2}-{\mathrm e}^{\frac {y +1}{x +2}} x^{3} b_{2}+{\mathrm e}^{\frac {y +1}{x +2}} x^{3} b_{3}+2 a_{1}-4 a_{2}-2 a_{3}-4 b_{1}+8 b_{2}+4 b_{3}-6 \,{\mathrm e}^{\frac {2 y +2}{x +2}} x^{2} a_{3}-12 \,{\mathrm e}^{\frac {2 y +2}{x +2}} x a_{3}+x y a_{1}-2 x y a_{2}-x y a_{3}-5 \,{\mathrm e}^{\frac {y +1}{x +2}} x^{2} a_{2}-4 \,{\mathrm e}^{\frac {y +1}{x +2}} x^{2} b_{2}+6 \,{\mathrm e}^{\frac {y +1}{x +2}} x^{2} b_{3}+2 \,{\mathrm e}^{\frac {y +1}{x +2}} y^{2} a_{3}-10 \,{\mathrm e}^{\frac {y +1}{x +2}} x a_{2}-8 \,{\mathrm e}^{\frac {y +1}{x +2}} x a_{3}-4 \,{\mathrm e}^{\frac {y +1}{x +2}} x b_{1}-4 \,{\mathrm e}^{\frac {y +1}{x +2}} x b_{2}+12 \,{\mathrm e}^{\frac {y +1}{x +2}} x b_{3}+2 \,{\mathrm e}^{\frac {y +1}{x +2}} y a_{1}-6 \,{\mathrm e}^{\frac {y +1}{x +2}} y a_{3}-4 \,{\mathrm e}^{\frac {y +1}{x +2}} y b_{3}-{\mathrm e}^{\frac {2 y +2}{x +2}} x^{3} a_{3}-2 x a_{2}-2 y a_{3}+8 x b_{2}-8 \,{\mathrm e}^{\frac {2 y +2}{x +2}} a_{3}-x^{2} b_{1}+2 x^{2} b_{2}+x^{2} b_{3}+x a_{1}-x a_{3}-4 x b_{1}+4 x b_{3}+2 y a_{1}-4 y a_{2}+2 \,{\mathrm e}^{\frac {y +1}{x +2}} a_{1}-8 \,{\mathrm e}^{\frac {y +1}{x +2}} a_{2}-8 \,{\mathrm e}^{\frac {y +1}{x +2}} a_{3}-4 \,{\mathrm e}^{\frac {y +1}{x +2}} b_{1}+8 \,{\mathrm e}^{\frac {y +1}{x +2}} b_{3} = 0
\end{equation}
Looking at the above PDE shows the following are all the terms with
\(\{x, y\}\) in them.
\[
\left \{x, y, {\mathrm e}^{\frac {y +1}{x +2}}, {\mathrm e}^{\frac {2 y +2}{x +2}}\right \}
\]
The
following substitution is now made to be able to collect on all terms with
\(\{x, y\}\) in them
\[
\left \{x = v_{1}, y = v_{2}, {\mathrm e}^{\frac {y +1}{x +2}} = v_{3}, {\mathrm e}^{\frac {2 y +2}{x +2}} = v_{4}\right \}
\]
The above
PDE (6E) now becomes
\begin{equation}
\tag{7E} -v_{3} v_{1}^{3} a_{2}+v_{3} v_{1}^{2} v_{2} a_{2}-v_{4} v_{1}^{3} a_{3}-2 v_{3} v_{1}^{2} v_{2} a_{3}+v_{3} v_{1} v_{2}^{2} a_{3}-v_{3} v_{1}^{3} b_{2}+v_{3} v_{1}^{3} b_{3}-v_{3} v_{1}^{2} v_{2} b_{3}+v_{3} v_{1} v_{2} a_{1}-5 v_{3} v_{1}^{2} a_{2}+2 v_{3} v_{1} v_{2} a_{2}-2 v_{3} v_{1}^{2} a_{3}-6 v_{4} v_{1}^{2} a_{3}-7 v_{3} v_{1} v_{2} a_{3}+2 v_{3} v_{2}^{2} a_{3}-v_{3} v_{1}^{2} b_{1}-4 v_{3} v_{1}^{2} b_{2}+6 v_{3} v_{1}^{2} b_{3}-4 v_{3} v_{1} v_{2} b_{3}+v_{1} v_{2} a_{1}+v_{3} v_{1} a_{1}+2 v_{3} v_{2} a_{1}-2 v_{1} v_{2} a_{2}-10 v_{3} v_{1} a_{2}-v_{1} v_{2} a_{3}-8 v_{3} v_{1} a_{3}-12 v_{4} v_{1} a_{3}-6 v_{3} v_{2} a_{3}-v_{1}^{2} b_{1}-4 v_{3} v_{1} b_{1}+2 v_{1}^{2} b_{2}-4 v_{3} v_{1} b_{2}+v_{1}^{2} b_{3}+12 v_{3} v_{1} b_{3}-4 v_{3} v_{2} b_{3}+v_{1} a_{1}+2 v_{2} a_{1}+2 v_{3} a_{1}-2 v_{1} a_{2}-4 v_{2} a_{2}-8 v_{3} a_{2}-v_{1} a_{3}-2 v_{2} a_{3}-8 v_{3} a_{3}-8 v_{4} a_{3}-4 v_{1} b_{1}-4 v_{3} b_{1}+8 v_{1} b_{2}+4 v_{1} b_{3}+8 v_{3} b_{3}+2 a_{1}-4 a_{2}-2 a_{3}-4 b_{1}+8 b_{2}+4 b_{3} = 0
\end{equation}
Collecting the above on the terms
\(v_i\) introduced, and these are
\[
\{v_{1}, v_{2}, v_{3}, v_{4}\}
\]
Equation
(7E) now becomes
\begin{equation}
\tag{8E} \left (-a_{2}-b_{2}+b_{3}\right ) v_{1}^{3} v_{3}-v_{4} v_{1}^{3} a_{3}+\left (a_{2}-2 a_{3}-b_{3}\right ) v_{1}^{2} v_{2} v_{3}+\left (-5 a_{2}-2 a_{3}-b_{1}-4 b_{2}+6 b_{3}\right ) v_{1}^{2} v_{3}-6 v_{4} v_{1}^{2} a_{3}+\left (-b_{1}+2 b_{2}+b_{3}\right ) v_{1}^{2}+v_{3} v_{1} v_{2}^{2} a_{3}+\left (a_{1}+2 a_{2}-7 a_{3}-4 b_{3}\right ) v_{1} v_{2} v_{3}+\left (a_{1}-2 a_{2}-a_{3}\right ) v_{1} v_{2}+\left (a_{1}-10 a_{2}-8 a_{3}-4 b_{1}-4 b_{2}+12 b_{3}\right ) v_{1} v_{3}-12 v_{4} v_{1} a_{3}+\left (a_{1}-2 a_{2}-a_{3}-4 b_{1}+8 b_{2}+4 b_{3}\right ) v_{1}+2 v_{3} v_{2}^{2} a_{3}+\left (2 a_{1}-6 a_{3}-4 b_{3}\right ) v_{2} v_{3}+\left (2 a_{1}-4 a_{2}-2 a_{3}\right ) v_{2}+\left (2 a_{1}-8 a_{2}-8 a_{3}-4 b_{1}+8 b_{3}\right ) v_{3}-8 v_{4} a_{3}+2 a_{1}-4 a_{2}-2 a_{3}-4 b_{1}+8 b_{2}+4 b_{3} = 0
\end{equation}
Setting each coefficients in (8E) to zero gives the following equations to solve
\begin{align*} a_{3}&=0\\ -12 a_{3}&=0\\ -8 a_{3}&=0\\ -6 a_{3}&=0\\ -a_{3}&=0\\ 2 a_{3}&=0\\ a_{1}-2 a_{2}-a_{3}&=0\\ 2 a_{1}-4 a_{2}-2 a_{3}&=0\\ 2 a_{1}-6 a_{3}-4 b_{3}&=0\\ -a_{2}-b_{2}+b_{3}&=0\\ a_{2}-2 a_{3}-b_{3}&=0\\ -b_{1}+2 b_{2}+b_{3}&=0\\ a_{1}+2 a_{2}-7 a_{3}-4 b_{3}&=0\\ 2 a_{1}-8 a_{2}-8 a_{3}-4 b_{1}+8 b_{3}&=0\\ -5 a_{2}-2 a_{3}-b_{1}-4 b_{2}+6 b_{3}&=0\\ a_{1}-10 a_{2}-8 a_{3}-4 b_{1}-4 b_{2}+12 b_{3}&=0\\ a_{1}-2 a_{2}-a_{3}-4 b_{1}+8 b_{2}+4 b_{3}&=0\\ 2 a_{1}-4 a_{2}-2 a_{3}-4 b_{1}+8 b_{2}+4 b_{3}&=0 \end{align*}
Solving the above equations for the unknowns gives
\begin{align*} a_{1}&=2 b_{3}\\ a_{2}&=b_{3}\\ a_{3}&=0\\ b_{1}&=b_{3}\\ 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 +2 \\
\eta &= y +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 \\ &= y +1 - \left (\frac {{\mathrm e}^{\frac {y +1}{x +2}} x +2 \,{\mathrm e}^{\frac {y +1}{x +2}}+y +1}{x +2}\right ) \left (x +2\right ) \\ &= \left (-x -2\right ) {\mathrm e}^{\frac {y +1}{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}{\left (-x -2\right ) {\mathrm e}^{\frac {y +1}{x +2}}}} dy \end{align*}
Which results in
\begin{align*} S&= {\mathrm e}^{-\frac {y +1}{x +2}} \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 {{\mathrm e}^{\frac {y +1}{x +2}} x +2 \,{\mathrm e}^{\frac {y +1}{x +2}}+y +1}{x +2} \end{align*}
Evaluating all the partial derivatives gives
\begin{align*} R_{x} &= 1\\ R_{y} &= 0\\ S_{x} &= \frac {{\mathrm e}^{\frac {-y -1}{x +2}} \left (y +1\right )}{\left (x +2\right )^{2}}\\ S_{y} &= -\frac {{\mathrm e}^{\frac {-y -1}{x +2}}}{x +2} \end{align*}
Substituting all the above in (2) and simplifying gives the ode in canonical coordinates.
\begin{align*} \frac {dS}{dR} &= -\frac {1}{x +2}\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 +2} \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 +2}\, dR}\\ S \left (R \right ) &= -\ln \left (R +2\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*} {\mathrm e}^{\frac {-y-1}{x +2}} = -\ln \left (x +2\right )+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 \((R,S)\)
\( \frac {dy}{dx} = \frac {{\mathrm e}^{\frac {y +1}{x +2}} x +2 \,{\mathrm e}^{\frac {y +1}{x +2}}+y +1}{x +2}\)
\( \frac {d S}{d R} = -\frac {1}{R +2}\)
\(\!\begin {aligned} R&= x\\ S&= {\mathrm e}^{\frac {-y -1}{x +2}} \end {aligned} \)
Solving for \(y\) gives
\begin{align*}
y &= -\ln \left (-\ln \left (x +2\right )+c_2 \right ) x -2 \ln \left (-\ln \left (x +2\right )+c_2 \right )-1 \\
\end{align*}
Figure 2.4: Slope field \(y^{\prime } = \frac {y+1}{x +2}+{\mathrm e}^{\frac {y+1}{x +2}}\) Summary of solutions found
\begin{align*}
y &= -\ln \left (-\ln \left (x +2\right )+c_2 \right ) x -2 \ln \left (-\ln \left (x +2\right )+c_2 \right )-1 \\
\end{align*}
✓ Time used: 0.006 (sec). Leaf size: 23 ode := diff ( y ( x ), x ) = (1+y(x))/(x+2)+exp((1+y(x))/(x+2));
dsolve ( ode , y ( x ), singsol=all);
\[
y = -1+\left (-x -2\right ) \ln \left (-\ln \left (x +2\right )-c_1 \right )
\]
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 C
trying homogeneous types:
trying homogeneous D
<- homogeneous successful
<- homogeneous successful
Maple step by step
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=\frac {y \left (x \right )+1}{x +2}+{\mathrm e}^{\frac {y \left (x \right )+1}{x +2}} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=\frac {y \left (x \right )+1}{x +2}+{\mathrm e}^{\frac {y \left (x \right )+1}{x +2}} \end {array} \]
✓ Time used: 0.761 (sec). Leaf size: 22 ode = D [ y [ x ], x ]==( y [ x ]+1)/( x +2)+ Exp [( y [ x ]+1)/( x +2)]; ic ={}; DSolve [{ ode , ic }, y [ x ], x , IncludeSingularSolutions -> True ] \begin{align*} y(x)&\to -1-((x+2) \log (-\log (x+2)+c_1)) \end{align*}
✗ from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(-exp((y(x) + 1)/(x + 2)) + Derivative(y(x), x) - (y(x) + 1)/(x + 2),0)
ics = {}
dsolve ( ode , func = y ( x ), ics = ics ) TypeError : argument of type Mul is not iterable