54.3.74 problem 1088

Internal problem ID [12369]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1088
Date solved : Friday, October 03, 2025 at 03:18:43 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 4 y^{\prime \prime }+4 y^{\prime } \tan \left (x \right )-\left (5 \tan \left (x \right )^{2}+2\right ) y&=0 \end{align*}

✓ Maple. Time used: 0.074 (sec). Leaf size: 29

ode:=4*diff(diff(y(x),x),x)+4*diff(y(x),x)*tan(x)-(5*tan(x)^2+2)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {i \sin \left (x \right ) \cos \left (x \right ) c_2 -\ln \left (\sin \left (x \right )+i \cos \left (x \right )\right ) c_2 +c_1}{\sqrt {\cos \left (x \right )}} \]

✓ Mathematica. Time used: 0.09 (sec). Leaf size: 97

ode=(-2 - 5*Tan[x]^2)*y[x] + 4*Tan[x]*D[y[x],x] + 4*D[y[x],{x,2}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {3 (-1)^{7/8} c_2 \text {arcsinh}\left (\frac {(1+i) \sqrt [4]{-\cos ^4(x)}}{\sqrt {2}}\right )+3 \sqrt [8]{-1} c_2 \sqrt [4]{-\cos ^4(x)} \sqrt {1+i \sqrt {-\cos ^4(x)}}-2 (-1)^{7/8} c_1}{2 \sqrt [8]{-\cos ^4(x)}} \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((-5*tan(x)**2 - 2)*y(x) + 4*tan(x)*Derivative(y(x), x) + 4*Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False