23.2.16 problem 16
Internal
problem
ID
[5371]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Part
II.
Chapter
2.
THE
DIFFERENTIAL
EQUATION
IS
OF
FIRST
ORDER
AND
OF
SECOND
OR
HIGHER
DEGREE,
page
278
Problem
number
:
16
Date
solved
:
Tuesday, September 30, 2025 at 12:38:07 PM
CAS
classification
:
[_quadrature]
\begin{align*} {y^{\prime }}^{2}&=a^{2} y^{n} \end{align*}
✓ Maple. Time used: 0.437 (sec). Leaf size: 72
ode:=diff(y(x),x)^2 = a^2*y(x)^n;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= 4^{\frac {1}{n -2}} \left (-\frac {1}{a \left (-c_1 +x \right ) \left (n -2\right )}\right )^{\frac {2}{n -2}} \\
y &= 4^{\frac {1}{n -2}} \left (\frac {1}{a \left (-c_1 +x \right ) \left (n -2\right )}\right )^{\frac {2}{n -2}} \\
\end{align*}
✓ Mathematica. Time used: 1.091 (sec). Leaf size: 77
ode=(D[y[x],x])^2==a^2*y[x]^n;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to 2^{\frac {2}{n-2}} (-((n-2) (a x+c_1))){}^{-\frac {2}{n-2}}\\ y(x)&\to 2^{\frac {2}{n-2}} ((n-2) (a x-c_1)){}^{-\frac {2}{n-2}}\\ y(x)&\to 0^{\frac {1}{n}} \end{align*}
✓ Sympy. Time used: 58.361 (sec). Leaf size: 299
from sympy import *
x = symbols("x")
a = symbols("a")
n = symbols("n")
y = Function("y")
ode = Eq(-a**2*y(x)**n + Derivative(y(x), x)**2,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = \begin {cases} 2^{\frac {2}{n - 2}} \left (C_{1}^{2} n^{2} - 4 C_{1}^{2} n + 4 C_{1}^{2} - 2 C_{1} a n^{2} x + 8 C_{1} a n x - 8 C_{1} a x + a^{2} n^{2} x^{2} - 4 a^{2} n x^{2} + 4 a^{2} x^{2}\right )^{- \frac {1}{n - 2}} & \text {for}\: n \neq 2 \\\text {NaN} & \text {otherwise} \end {cases}, \ y{\left (x \right )} = \begin {cases} e^{- \frac {2 W\left (- \frac {C_{1} n}{2} + C_{1} + \frac {a n x}{2} - a x\right )}{n - 2}} & \text {for}\: n = 2 \\\text {NaN} & \text {otherwise} \end {cases}, \ y{\left (x \right )} = \begin {cases} e^{- \frac {2 W\left (\frac {C_{1} n}{2} - C_{1} - \frac {a n x}{2} + a x\right )}{n - 2}} & \text {for}\: n = 2 \\\text {NaN} & \text {otherwise} \end {cases}, \ y{\left (x \right )} = \begin {cases} 2^{\frac {2}{n - 2}} \left (C_{1}^{2} n^{2} - 4 C_{1}^{2} n + 4 C_{1}^{2} + 2 C_{1} a n^{2} x - 8 C_{1} a n x + 8 C_{1} a x + a^{2} n^{2} x^{2} - 4 a^{2} n x^{2} + 4 a^{2} x^{2}\right )^{- \frac {1}{n - 2}} & \text {for}\: n \neq 2 \\\text {NaN} & \text {otherwise} \end {cases}, \ y{\left (x \right )} = \begin {cases} e^{- \frac {2 W\left (- \frac {C_{1} n}{2} + C_{1} - \frac {a n x}{2} + a x\right )}{n - 2}} & \text {for}\: n = 2 \\\text {NaN} & \text {otherwise} \end {cases}, \ y{\left (x \right )} = \begin {cases} e^{- \frac {2 W\left (\frac {C_{1} n}{2} - C_{1} + \frac {a n x}{2} - a x\right )}{n - 2}} & \text {for}\: n = 2 \\\text {NaN} & \text {otherwise} \end {cases}\right ]
\]