60.3.68 problem 1079

Internal problem ID [11064]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1079
Date solved : Sunday, March 30, 2025 at 07:41:28 PM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

\begin{align*} y^{\prime \prime }-\frac {a f^{\prime }\left (x \right ) y^{\prime }}{f \left (x \right )}+b f \left (x \right )^{2 a} y&=0 \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 33

ode:=diff(diff(y(x),x),x)-a*diff(f(x),x)/f(x)*diff(y(x),x)+b*f(x)^(2*a)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = c_1 \,{\mathrm e}^{i \sqrt {b}\, \int f \left (x \right )^{a}d x}+c_2 \,{\mathrm e}^{-i \sqrt {b}\, \int f \left (x \right )^{a}d x} \]

✓ Mathematica. Time used: 0.512 (sec). Leaf size: 139

ode=b*f[x]^(2*a)*y[x] - (a*Derivative[1][f][x]*D[y[x],x])/f[x] + D[y[x],{x,2}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to \frac {1}{2} \left (\exp \left (\int _1^x-i \sqrt {b} f(K[1])^adK[1]+c_2\right )-2 c_1 \exp \left (-\int _1^x-i \sqrt {b} f(K[1])^adK[1]-c_2\right )\right ) \\ y(x)\to \frac {1}{2} \left (\exp \left (\int _1^xi \sqrt {b} f(K[2])^adK[2]+c_2\right )-2 c_1 \exp \left (-\int _1^xi \sqrt {b} f(K[2])^adK[2]-c_2\right )\right ) \\ \end{align*}

✗ Sympy

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

TypeError : cannot determine truth value of Relational