Internal problem ID [3587]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 29
Problem number: 857.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_homogeneous, class G]]
Solve \begin {gather*} \boxed {x \left (y^{\prime }\right )^{2}+y y^{\prime }+x^{3}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.112 (sec). Leaf size: 337
dsolve(x*diff(y(x),x)^2+y(x)*diff(y(x),x)+x^3 = 0,y(x), singsol=all)
\begin{align*} \int _{\textit {\_b}}^{x}-\frac {-y \relax (x )+\sqrt {-4 \textit {\_a}^{4}+y \relax (x )^{2}}}{\textit {\_a} \left (-5 y \relax (x )+\sqrt {-4 \textit {\_a}^{4}+y \relax (x )^{2}}\right )}d \textit {\_a} +\int _{}^{y \relax (x )}-\frac {2 \left (8 \left (\int _{\textit {\_b}}^{x}\frac {\textit {\_a}^{3}}{\left (-5 \textit {\_f} +\sqrt {-4 \textit {\_a}^{4}+\textit {\_f}^{2}}\right )^{2} \sqrt {-4 \textit {\_a}^{4}+\textit {\_f}^{2}}}d \textit {\_a} \right ) \sqrt {-4 x^{4}+\textit {\_f}^{2}}-40 \left (\int _{\textit {\_b}}^{x}\frac {\textit {\_a}^{3}}{\left (-5 \textit {\_f} +\sqrt {-4 \textit {\_a}^{4}+\textit {\_f}^{2}}\right )^{2} \sqrt {-4 \textit {\_a}^{4}+\textit {\_f}^{2}}}d \textit {\_a} \right ) \textit {\_f} -1\right )}{-5 \textit {\_f} +\sqrt {-4 x^{4}+\textit {\_f}^{2}}}d \textit {\_f} +c_{1} = 0 \\ \int _{\textit {\_b}}^{x}-\frac {y \relax (x )+\sqrt {-4 \textit {\_a}^{4}+y \relax (x )^{2}}}{\left (\sqrt {-4 \textit {\_a}^{4}+y \relax (x )^{2}}+5 y \relax (x )\right ) \textit {\_a}}d \textit {\_a} +\int _{}^{y \relax (x )}\frac {16 \left (\int _{\textit {\_b}}^{x}\frac {\textit {\_a}^{3}}{\left (\sqrt {-4 \textit {\_a}^{4}+\textit {\_f}^{2}}+5 \textit {\_f} \right )^{2} \sqrt {-4 \textit {\_a}^{4}+\textit {\_f}^{2}}}d \textit {\_a} \right ) \sqrt {-4 x^{4}+\textit {\_f}^{2}}+80 \left (\int _{\textit {\_b}}^{x}\frac {\textit {\_a}^{3}}{\left (\sqrt {-4 \textit {\_a}^{4}+\textit {\_f}^{2}}+5 \textit {\_f} \right )^{2} \sqrt {-4 \textit {\_a}^{4}+\textit {\_f}^{2}}}d \textit {\_a} \right ) \textit {\_f} -2}{\sqrt {-4 x^{4}+\textit {\_f}^{2}}+5 \textit {\_f}}d \textit {\_f} +c_{1} = 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 25.041 (sec). Leaf size: 107
DSolve[x (y'[x])^2+y[x] y'[x]+x^3==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to x^2 \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{5 K[2]+\sqrt {K[2]^2-4}}dK[2]\&\right ]\left [\int _1^x-\frac {1}{2 K[3]}dK[3]+c_1\right ] \\ y(x)\to x^2 \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{\sqrt {K[4]^2-4}-5 K[4]}dK[4]\&\right ]\left [\int _1^x\frac {1}{2 K[5]}dK[5]+c_1\right ] \\ \end{align*}