2.2.2 Problem 2

Internal problem ID [9124]
Book : Second order enumerated odes
Section : section 2
Problem number : 2
Date solved : Friday, April 25, 2025 at 05:54:00 PM
CAS classification : [_Liouville, [_2nd_order, _reducible, _mu_xy]]

Solved as second nonlinear ode solved by Mainardi Lioville method

Time used: 0.365 (sec)

Solve

$$\begin{array}{rl}{y}^{\prime \prime }+\sin \left(x\right)y^{\prime }+y{y^{\prime }}^2& =0\end{array}$$

The ode has the Liouville form given by

$$\begin{array}{}\text{(1A)}& y^{\prime \prime }+f(x)y^{\prime }+g(y){y^{\prime }}^2& =0\end{array}$$

Where in this problem

$$\begin{array}{rl}f(x)& =\sin \left(x\right)\\ g(y)& =y\end{array}$$

Dividing through by $y^{\prime }$ then Eq (1A) becomes

$$\begin{array}{}\text{(2A)}& \frac{y^{\prime \prime }}{y^{\prime }}+f+g{y}^{\prime }& =0\end{array}$$

But the first term in Eq (2A) can be written as

$$\begin{array}{}\text{(3A)}& \frac{y^{\prime \prime }}{y^{\prime }}& =\frac{d}{dx}\ln \left(y^{\prime }\right)\end{array}$$

And the last term in Eq (2A) can be written as

$$\begin{array}{rl}g\frac{dy}{dx}& =(\frac{d}{dy}\int gdy)\frac{dy}{dx}\\ \text{(4A)}& & =\frac{d}{dx}\int gdy\end{array}$$

Substituting (3A,4A) back into (2A) gives

$$\begin{array}{}\text{(5A)}& \frac{d}{dx}\ln \left(y^{\prime }\right)+\frac{d}{dx}\int gdy& =-f\end{array}$$

Integrating the above w.r.t. $x$ gives

$$\begin{array}{rl}\ln \left(y^{\prime }\right)+\int gdy& =-\int fdx+c_1\end{array}$$

Where $c_1$ is arbitrary constant. Taking the exponential of the above gives

$$\begin{array}{}\text{(6A)}& y^{\prime }& =c_2{e}^{\int -gdy}{e}^{\int -fdx}\end{array}$$

Where $c_2$ is a new arbitrary constant. But since $g=y$ and $f=\sin \left(x\right)$, then

$$\begin{array}{rl}\int -gdy& =\int -ydy\\ & =-\frac{y^2}{2}\\ \int -fdx& =\int -\sin \left(x\right)dx\\ & =\cos \left(x\right)\end{array}$$

Substituting the above into Eq(6A) gives

$$y^{\prime }=c_2{\mathrm{e}}^{-\frac{y^2}{2}}{\mathrm{e}}^{\cos \left(x\right)}$$

Which is now solved as first order separable ode. The ode

$$\begin{array}{}\text{(1)}& y^{\prime }=c_2{\mathrm{e}}^{-\frac{y^2}{2}}{\mathrm{e}}^{\cos \left(x\right)}\end{array}$$

is separable as it can be written as

$$\begin{array}{rl}{y}^{\prime }& =c_2{\mathrm{e}}^{-\frac{y^2}{2}}{\mathrm{e}}^{\cos \left(x\right)}\\ & =f(x)g(y)\end{array}$$

Where

$$\begin{array}{rl}f(x)& ={\mathrm{e}}^{\cos \left(x\right)}{c}_2\\ g(y)& ={\mathrm{e}}^{-\frac{y^2}{2}}\end{array}$$

Integrating gives

$$\begin{array}{rl}\int \frac{1}{g(y)}dy& =\int f(x)dx\\ \int {\mathrm{e}}^{\frac{y^2}{2}}dy& =\int {\mathrm{e}}^{\cos \left(x\right)}{c}_{2}dx\end{array}$$

$$-\frac{i\sqrt{\pi }\sqrt{2}\mathrm{erf}\left(\frac{i\sqrt{2}y}{2}\right)}{2}=\int {\mathrm{e}}^{\cos \left(x\right)}{c}_{2}dx+2c_3$$

Will add steps showing solving for IC soon.

Summary of solutions found

$$\begin{array}{rl}-\frac{i\sqrt{\pi }\sqrt{2}\mathrm{erf}\left(\frac{i\sqrt{2}y}{2}\right)}{2}& =\int {\mathrm{e}}^{\cos \left(x\right)}{c}_{2}dx+2c_3\end{array}$$

✓ Maple. Time used: 0.005 (sec). Leaf size: 35

ode:=diff(diff(y(x),x),x)+diff(y(x),x)*sin(x)+y(x)*diff(y(x),x)^2 = 0; 
dsolve(ode,y(x), singsol=all);
 

$$y=-i\mathrm{RootOf}(i\sqrt{2}{c}_1\int {\mathrm{e}}^{\cos \left(x\right)}dx+i\sqrt{2}{c}_2-\mathrm{erf}\left(\mathit{\text{\_Z}}\right)\sqrt{\pi })\sqrt{2}$$

Maple trace

Methods for second order ODEs: 
--- Trying classification methods --- 
trying 2nd order Liouville 
<- 2nd_order Liouville successful
 

✓ Mathematica. Time used: 66.642 (sec). Leaf size: 76

ode=D[y[x],{x,2}]+Sin[x]*D[y[x],x]+y[x]*(D[y[x],x])^2==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

$$\begin{array}{r}y(x)\to -i\sqrt{2}{\text{erf}}^{-1}\left(i\sqrt{\frac{2}{\pi }}({\int }_1^x-e^{\cos (K[2])}{c}_{1}dK[2]+c_2)\right)\\ y(x)\to -i\sqrt{2}{\text{erf}}^{-1}\left(i\sqrt{\frac{2}{\pi }}{c}_2\right)\end{array}$$

✗ Sympy

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

NotImplementedError : The given ODE (sqrt(-4*y(x)*Derivative(y(x), (x, 2)) + sin(x)**2) + sin(x))/(2*y(x)) + Derivative(y(x), x) cannot be solved by the factorable group method