77.1.48 problem 67 (page 109)

Internal problem ID [17938]
Book : V.V. Stepanov, A course of differential equations (in Russian), GIFML. Moscow (1958)
Section : All content
Problem number : 67 (page 109)
Date solved : Tuesday, January 28, 2025 at 11:14:09 AM
CAS classification : [_quadrature]

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

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 32

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

\begin{align*} y &= \frac {x^{3}}{3}+c_{1} \\ y &= \frac {1}{-x +c_{1}} \\ y &= {\mathrm e}^{\frac {x^{2}}{2}} c_{1} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.114 (sec). Leaf size: 48

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

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