90.3.16 problem 16

Internal problem ID [25080]
Book : Ordinary Differential Equations. By William Adkins and Mark G Davidson. Springer. NY. 2010. ISBN 978-1-4614-3617-1
Section : Chapter 1. First order differential equations. Exercises at page 41
Problem number : 16
Date solved : Thursday, October 02, 2025 at 11:49:32 PM
CAS classification : [_separable]

\begin{align*} y^{\prime }&=t^{m} y^{n} \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 41

ode:=diff(y(t),t) = t^m*y(t)^n; 
dsolve(ode,y(t), singsol=all);
 

\[ y = \left (-\frac {t^{m +1} n -c_1 m -t^{m +1}-c_1}{m +1}\right )^{-\frac {1}{n -1}} \]

✓ Mathematica. Time used: 4.3 (sec). Leaf size: 44

ode=D[y[t],{t,1}] ==t^m*y[t]^n; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to \left (-\frac {(n-1) \left (t^{m+1}+c_1 m+c_1\right )}{m+1}\right ){}^{\frac {1}{1-n}}\\ y(t)&\to 0^{\frac {1}{n}} \end{align*}

✓ Sympy. Time used: 0.656 (sec). Leaf size: 126

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

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