29.18.35 problem 513

Internal problem ID [5109]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 18
Problem number : 513
Date solved : Monday, January 27, 2025 at 10:11:47 AM
CAS classification : [[_homogeneous, `class G`], _rational, _Bernoulli]

\begin{align*} x y y^{\prime }&=a \,x^{n}+b y^{2} \end{align*}

✓ Solution by Maple

Time used: 0.010 (sec). Leaf size: 84

dsolve(x*y(x)*diff(y(x),x) = a*x^n+b*y(x)^2,y(x), singsol=all)
 

\begin{align*} y \left (x \right ) &= \frac {\sqrt {-4 \left (-c_{1} \left (b -\frac {n}{2}\right ) x^{2 b}+a \,x^{n}\right ) \left (b -\frac {n}{2}\right )}}{2 b -n} \\ y \left (x \right ) &= -\frac {\sqrt {-4 \left (-c_{1} \left (b -\frac {n}{2}\right ) x^{2 b}+a \,x^{n}\right ) \left (b -\frac {n}{2}\right )}}{2 b -n} \\ \end{align*}

✓ Solution by Mathematica

Time used: 4.793 (sec). Leaf size: 86

DSolve[x y[x] D[y[x],x]==a x^n+b y[x]^2,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to -\frac {\sqrt {-2 a x^n+c_1 (2 b-n) x^{2 b}}}{\sqrt {2 b-n}} \\ y(x)\to \frac {\sqrt {-2 a x^n+c_1 (2 b-n) x^{2 b}}}{\sqrt {2 b-n}} \\ \end{align*}