29.7.16 problem 191

Internal problem ID [4791]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 7
Problem number : 191
Date solved : Monday, January 27, 2025 at 09:38:06 AM
CAS classification : [[_homogeneous, `class G`], _rational, _Bernoulli]

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

✓ Solution by Maple

Time used: 0.014 (sec). Leaf size: 47

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

\[ y \left (x \right ) = 2^{\frac {1}{-1+k}} x^{-\frac {2 k}{-1+k}} \left (\frac {-a \left (-1+k \right ) x^{2}+2 c_{1}}{x^{2}}\right )^{-\frac {1}{-1+k}} \]

✓ Solution by Mathematica

Time used: 15.959 (sec). Leaf size: 45

DSolve[x D[y[x],x]+2 y[x]==a x^(2 k)y[x]^k,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \left (\frac {1}{2} a x^{2 k}-\frac {1}{2} a k x^{2 k}+c_1 x^{2 k-2}\right ){}^{\frac {1}{1-k}} \]