29.34.14 problem 1016

Internal problem ID [5585]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 34
Problem number : 1016
Date solved : Monday, January 27, 2025 at 12:12:46 PM
CAS classification : [_quadrature]

\begin{align*} {y^{\prime }}^{3}&=a \,x^{n} \end{align*}

✓ Solution by Maple

Time used: 0.063 (sec). Leaf size: 98

dsolve(diff(y(x),x)^3 = a*x^n,y(x), singsol=all)
 

\begin{align*} y \left (x \right ) &= \frac {3 x \left (a \,x^{n}\right )^{{1}/{3}}+c_{1} \left (n +3\right )}{n +3} \\ y \left (x \right ) &= \frac {\left (-3 i \sqrt {3}\, x -3 x \right ) \left (a \,x^{n}\right )^{{1}/{3}}+2 c_{1} \left (n +3\right )}{2 n +6} \\ y \left (x \right ) &= \frac {\left (3 i \sqrt {3}\, x -3 x \right ) \left (a \,x^{n}\right )^{{1}/{3}}+2 c_{1} \left (n +3\right )}{2 n +6} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.008 (sec). Leaf size: 95

DSolve[(D[y[x],x])^3 ==a x^n,y[x],x,IncludeSingularSolutions -> True]
 

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