29.14.5 problem 385

Internal problem ID [4984]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 14
Problem number : 385
Date solved : Sunday, March 30, 2025 at 04:28:15 AM
CAS classification : [[_homogeneous, `class G`], _Riccati]

\begin{align*} x^{n} y^{\prime }&=a^{2} x^{2 n -2}+b^{2} y^{2} \end{align*}

✓ Maple. Time used: 0.010 (sec). Leaf size: 67

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

\[ y = \frac {x^{n -1} \left (n -1+\tan \left (\frac {\sqrt {4 a^{2} b^{2}-n^{2}+2 n -1}\, \left (\ln \left (x \right )-c_1 \right )}{2}\right ) \sqrt {4 a^{2} b^{2}-n^{2}+2 n -1}\right )}{2 b^{2}} \]

✓ Mathematica. Time used: 0.538 (sec). Leaf size: 162

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

\begin{align*} y(x)\to \frac {x^{n-1} \left (\left (-a b \sqrt {\frac {(n-1)^2}{a^2 b^2}-4}+n-1\right ) x^{a b \sqrt {\frac {(n-1)^2}{a^2 b^2}-4}}+c_1 \left (a b \sqrt {\frac {(n-1)^2}{a^2 b^2}-4}+n-1\right )\right )}{2 b^2 \left (x^{a b \sqrt {\frac {(n-1)^2}{a^2 b^2}-4}}+c_1\right )} \\ y(x)\to \frac {x^{n-1} \left (a b \sqrt {\frac {(n-1)^2}{a^2 b^2}-4}+n-1\right )}{2 b^2} \\ \end{align*}

✗ Sympy

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

NotImplementedError : The given ODE Derivative(y(x), x) - (a**2*x**(2*n - 2) + b**2*y(x)**2)/x**n cannot be solved by the factorable group method