23.1.196 problem 194

Internal problem ID [4803]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 1. THE DIFFERENTIAL EQUATION IS OF FIRST ORDER AND OF FIRST DEGREE, page 223
Problem number : 194
Date solved : Tuesday, September 30, 2025 at 08:41:10 AM
CAS classification : [[_homogeneous, `class G`], _rational, _Bernoulli]

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

✓ Maple. Time used: 0.021 (sec). Leaf size: 41

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

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

✓ Mathematica. Time used: 15.48 (sec). Leaf size: 45

ode=x*D[y[x],x]+2*y[x]==a*x^(2*k)*y[x]^k; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} 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}} \end{align*}

✓ Sympy. Time used: 0.873 (sec). Leaf size: 41

from sympy import * 
x = symbols("x") 
a = symbols("a") 
k = symbols("k") 
y = Function("y") 
ode = Eq(-a*x**(2*k)*y(x)**k + x*Derivative(y(x), x) + 2*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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