29.35.19 problem 1052

Internal problem ID [5617]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 35
Problem number : 1052
Date solved : Monday, January 27, 2025 at 12:26:59 PM
CAS classification : [_quadrature]

\begin{align*} {y^{\prime }}^{3}-\left (x^{2}+x y^{2}+y^{4}\right ) {y^{\prime }}^{2}+x y^{2} \left (x^{2}+x y^{2}+y^{4}\right ) y^{\prime }-x^{3} y^{6}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 73

dsolve(diff(y(x),x)^3-(x^2+x*y(x)^2+y(x)^4)*diff(y(x),x)^2+x*y(x)^2*(x^2+x*y(x)^2+y(x)^4)*diff(y(x),x)-x^3*y(x)^6 = 0,y(x), singsol=all)
 

\begin{align*} y \left (x \right ) &= \frac {x^{3}}{3}+c_{1} \\ y \left (x \right ) &= \frac {1}{\left (-3 x +c_{1} \right )^{{1}/{3}}} \\ y \left (x \right ) &= -\frac {1+i \sqrt {3}}{2 \left (-3 x +c_{1} \right )^{{1}/{3}}} \\ y \left (x \right ) &= \frac {i \sqrt {3}-1}{2 \left (-3 x +c_{1} \right )^{{1}/{3}}} \\ y \left (x \right ) &= -\frac {2}{x^{2}-2 c_{1}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.251 (sec). Leaf size: 110

DSolve[(D[y[x],x])^3 -(x^2+x y[x]^2+ y[x]^4) (D[y[x],x])^2 +x y[x]^2(x^2 +x y[x]^2+ y[x]^4) D[y[x],x]-x^3 y[x]^6==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to -\frac {\sqrt [3]{-\frac {1}{3}}}{\sqrt [3]{-x-c_1}} \\ y(x)\to \frac {1}{\sqrt [3]{3} \sqrt [3]{-x-c_1}} \\ y(x)\to \frac {(-1)^{2/3}}{\sqrt [3]{3} \sqrt [3]{-x-c_1}} \\ y(x)\to \frac {x^3}{3}+c_1 \\ y(x)\to -\frac {2}{x^2+2 c_1} \\ y(x)\to 0 \\ \end{align*}