89.7.21 problem 21

Internal problem ID [24452]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 4. Additional topics on equations of first order and first degree. Exercises at page 61
Problem number : 21
Date solved : Thursday, October 02, 2025 at 10:36:50 PM
CAS classification : [[_homogeneous, `class G`], _rational, _Bernoulli]

\begin{align*} x y^{\prime }-y&=x^{k} y^{n} \end{align*}

✓ Maple. Time used: 0.020 (sec). Leaf size: 59

ode:=x*diff(y(x),x)-y(x) = x^k*y(x)^n; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \left (\frac {k \,x^{1-n} c_1 +n \,x^{1-n} c_1 -x^{k} n -c_1 \,x^{1-n}+x^{k}}{n -1+k}\right )^{-\frac {1}{n -1}} \]

✓ Mathematica. Time used: 5.015 (sec). Leaf size: 38

ode=x*D[y[x],x]-y[x]==x^k*y[x]^n; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \left (-\frac {(n-1) x^k}{k+n-1}+c_1 x^{1-n}\right ){}^{\frac {1}{1-n}} \end{align*}

✓ Sympy. Time used: 1.268 (sec). Leaf size: 207

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

\[ y{\left (x \right )} = \begin {cases} \left (\frac {C_{1} k x}{k e^{n \log {\left (x \right )}} + n e^{n \log {\left (x \right )}} - e^{n \log {\left (x \right )}}} + \frac {C_{1} n x}{k e^{n \log {\left (x \right )}} + n e^{n \log {\left (x \right )}} - e^{n \log {\left (x \right )}}} - \frac {C_{1} x}{k e^{n \log {\left (x \right )}} + n e^{n \log {\left (x \right )}} - e^{n \log {\left (x \right )}}} - \frac {n x^{k} e^{n \log {\left (x \right )}}}{k e^{n \log {\left (x \right )}} + n e^{n \log {\left (x \right )}} - e^{n \log {\left (x \right )}}} + \frac {x^{k} e^{n \log {\left (x \right )}}}{k e^{n \log {\left (x \right )}} + n e^{n \log {\left (x \right )}} - e^{n \log {\left (x \right )}}}\right )^{- \frac {1}{n - 1}} & \text {for}\: k + n \neq 1 \\\left (C_{1} x e^{- n \log {\left (x \right )}} - n x e^{- n \log {\left (x \right )}} \log {\left (x \right )} + x e^{- n \log {\left (x \right )}} \log {\left (x \right )}\right )^{- \frac {1}{n - 1}} & \text {otherwise} \end {cases} \]