54.3.67 problem 1078

Internal problem ID [12362]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1078
Date solved : Wednesday, October 01, 2025 at 01:43:52 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+f \left (x \right ) y^{\prime }+\left (\frac {f \left (x \right )^{2}}{4}+\frac {f^{\prime }\left (x \right )}{2}+a \right ) y&=0 \end{align*}

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

ode:=diff(diff(y(x),x),x)+f(x)*diff(y(x),x)+(1/4*f(x)^2+1/2*diff(f(x),x)+a)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = {\mathrm e}^{-\frac {\int f \left (x \right )d x}{2}} \left (c_1 \sinh \left (\sqrt {-a}\, x \right )+c_2 \cosh \left (\sqrt {-a}\, x \right )\right ) \]

✓ Mathematica. Time used: 0.074 (sec). Leaf size: 69

ode=y[x]*(a + f[x]^2/4 + Derivative[1][f][x]/2) + f[x]*D[y[x],x] + D[y[x],{x,2}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {\left (2 \sqrt {a} c_1-i c_2 e^{2 i \sqrt {a} x}\right ) \exp \left (-\frac {1}{2} \int _1^xf(K[1])dK[1]-i \sqrt {a} x\right )}{2 \sqrt {a}} \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
f = Function("f") 
ode = Eq((a + f(x)**2/4 + Derivative(f(x), x)/2)*y(x) + f(x)*Derivative(y(x), x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

TypeError : cannot determine truth value of Relational: x > _n + 2