12.1.16 problem 8(a)

Internal problem ID [1534]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 1, Introduction. Section 1.2 Page 14
Problem number : 8(a)
Date solved : Saturday, March 29, 2025 at 10:57:58 PM
CAS classification : [_quadrature]

\begin{align*} y^{\prime }&=a y^{\frac {a -1}{a}} \end{align*}

✓ Maple. Time used: 0.005 (sec). Leaf size: 9

ode:=diff(y(x),x) = a*y(x)^((a-1)/a); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \left (x +c_1 \right )^{a} \]

✓ Mathematica. Time used: 0.88 (sec). Leaf size: 28

ode=D[y[x],x] ==a*y[x]^( (a-1)/a); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to \left (x+\frac {c_1}{a}\right ){}^a \\ y(x)\to 0^{\frac {a}{a-1}} \\ \end{align*}

✓ Sympy. Time used: 0.256 (sec). Leaf size: 24

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(-a*y(x)**((a - 1)/a) + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ y{\left (x \right )} = \begin {cases} \left (\frac {C_{1}}{a} + x\right )^{a} & \text {for}\: \frac {1}{a} \neq 0 \\\text {NaN} & \text {otherwise} \end {cases}, \ y{\left (x \right )} = \begin {cases} C_{1} e^{a x} & \text {for}\: \frac {1}{a} = 0 \\\text {NaN} & \text {otherwise} \end {cases}\right ] \]