2.2.8 problem 9

Internal problem ID [8543]
Book : Second order enumerated odes
Section : section 2
Problem number : 9
Date solved : Sunday, November 10, 2024 at 09:12:30 PM
CAS classification : [_Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

Solve

\begin{align*} y^{\prime \prime }+\left (3+x \right ) y^{\prime }+\left (3+y^{2}\right ) {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 : 0.014 (sec)
Leaf size : 32

dsolve(diff(diff(y(x),x),x)+(x+3)*diff(y(x),x)+(y(x)^2+3)*diff(y(x),x)^2 = 0, 
       y(x),singsol=all)
 

\[ c_{1} \operatorname {erf}\left (\frac {\sqrt {2}\, \left (x +3\right )}{2}\right )-c_{2} +\int _{}^{y}{\mathrm e}^{\frac {\textit {\_a} \left (\textit {\_a}^{2}+9\right )}{3}}d \textit {\_a} = 0 \]

Mathematica DSolve solution

Solving time : 0.609 (sec)
Leaf size : 61

DSolve[{D[y[x],{x,2}]+(3+x)*D[y[x],x]+(3+y[x]^2)*(D[y[x],x])^2==0,{}}, 
       y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}e^{\frac {K[1]^3}{3}+3 K[1]}dK[1]\&\right ]\left [c_2-e^{9/2} \sqrt {\frac {\pi }{2}} c_1 \text {erf}\left (\frac {x+3}{\sqrt {2}}\right )\right ] \]