60.3.69 problem 1080

Internal problem ID [11065]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1080
Date solved : Thursday, March 13, 2025 at 08:23:55 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.128 (sec). Leaf size: 65

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

\[ y = {\mathrm e}^{\int \frac {\left (\left (-b f+a \right ) {\mathrm e}^{-2 b \left (\int fd x -c_{1} \right )}-b f-a \right ) {\mathrm e}^{2 b \left (\int fd x \right )}}{-{\mathrm e}^{2 b \left (\int fd x \right )}+{\mathrm e}^{2 c_{1} b}}d x} c_{2} \]

✓ Mathematica. Time used: 0.081 (sec). Leaf size: 47

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

\[ y(x)\to e^{a x} \left (c_1 \exp \left (b \int _1^xf(K[1])dK[1]\right )+c_2 \exp \left (-b \int _1^xf(K[2])dK[2]\right )\right ) \]

✗ Sympy

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

TypeError : cannot determine truth value of Relational