2.2.4 Problem 4

Internal problem ID [9127]
Book : Second order enumerated odes
Section : section 2
Problem number : 4
Date solved : Tuesday, January 28, 2025 at 04:00:07 PM
CAS classification : [_Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

Solve

\begin{align*} y^{\prime \prime }+\left (\sin \left (x \right )+2 x \right ) y^{\prime }+\cos \left (y\right ) y {y^{\prime }}^{2}&=0 \end{align*}

Maple step by step solution

Maple trace

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

Maple dsolve solution

Solving time : 0.005 (sec)
Leaf size : 34

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

\[ \int _{}^{y}{\mathrm e}^{\cos \left (\textit {\_a} \right )+\textit {\_a} \sin \left (\textit {\_a} \right )}d \textit {\_a} -c_{1} \left (\int {\mathrm e}^{-x^{2}+\cos \left (x \right )}d x \right )-c_{2} = 0 \]

Mathematica DSolve solution

Solving time : 1.353 (sec)
Leaf size : 68

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

\[ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\exp \left (-\int _1^{K[3]}-\cos (K[1]) K[1]dK[1]\right )dK[3]\&\right ]\left [\int _1^x-\exp \left (-\int _1^{K[4]}(2 K[2]+\sin (K[2]))dK[2]\right ) c_1dK[4]+c_2\right ] \]