Internal problem ID [4095]
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`]]
\[ \boxed {{y^{\prime }}^{2} x +y y^{\prime }=-x^{3}} \]
✓ Solution by Maple
Time used: 0.125 (sec). Leaf size: 272
dsolve(x*diff(y(x),x)^2+y(x)*diff(y(x),x)+x^3 = 0,y(x), singsol=all)
\begin{align*} -\left (\int _{\textit {\_b}}^{x}\frac {y \left (x \right )-\sqrt {-4 \textit {\_a}^{4}+y \left (x \right )^{2}}}{\textit {\_a} \left (5 y \left (x \right )-\sqrt {-4 \textit {\_a}^{4}+y \left (x \right )^{2}}\right )}d \textit {\_a} \right )-2 \left (\int _{}^{y \left (x \right )}\frac {1+\left (40 \textit {\_f} -8 \sqrt {-4 x^{4}+\textit {\_f}^{2}}\right ) \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 )}{5 \textit {\_f} -\sqrt {-4 x^{4}+\textit {\_f}^{2}}}d \textit {\_f} \right )+c_{1} &= 0 \\ -\left (\int _{\textit {\_b}}^{x}\frac {y \left (x \right )+\sqrt {-4 \textit {\_a}^{4}+y \left (x \right )^{2}}}{\left (\sqrt {-4 \textit {\_a}^{4}+y \left (x \right )^{2}}+5 y \left (x \right )\right ) \textit {\_a}}d \textit {\_a} \right )+2 \left (\int _{}^{y \left (x \right )}\frac {-1+8 \left (\sqrt {-4 x^{4}+\textit {\_f}^{2}}+5 \textit {\_f} \right ) \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}}+5 \textit {\_f}}d \textit {\_f} \right )+c_{1} &= 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.825 (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*}