29.23.31 problem 662

Internal problem ID [5253]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 23
Problem number : 662
Date solved : Monday, January 27, 2025 at 10:49:49 AM
CAS classification : [_separable]

\begin{align*} x^{2} y^{2} y^{\prime }+1-x +x^{3}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.013 (sec). Leaf size: 111

dsolve(x^2*y(x)^2*diff(y(x),x)+1-x+x^3 = 0,y(x), singsol=all)
 

\begin{align*} y \left (x \right ) &= \frac {2^{{2}/{3}} \left (-3 x^{2} \left (x^{3}-\frac {2 c_{1} x}{3}-2 x \ln \left (x \right )-2\right )\right )^{{1}/{3}}}{2 x} \\ y \left (x \right ) &= -\frac {2^{{2}/{3}} \left (-3 x^{2} \left (x^{3}-\frac {2 c_{1} x}{3}-2 x \ln \left (x \right )-2\right )\right )^{{1}/{3}} \left (1+i \sqrt {3}\right )}{4 x} \\ y \left (x \right ) &= \frac {2^{{2}/{3}} \left (-3 x^{2} \left (x^{3}-\frac {2 c_{1} x}{3}-2 x \ln \left (x \right )-2\right )\right )^{{1}/{3}} \left (i \sqrt {3}-1\right )}{4 x} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.426 (sec). Leaf size: 111

DSolve[x^2 y[x]^2 D[y[x],x]+1-x+x^3==0,y[x],x,IncludeSingularSolutions -> True]
 

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