54.3.163 problem 1177

Internal problem ID [12458]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1177
Date solved : Wednesday, October 01, 2025 at 01:45:18 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.006 (sec). Leaf size: 30

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

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

✓ Mathematica. Time used: 0.615 (sec). Leaf size: 208

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

\begin{align*} y(x)&\to \frac {\exp \left (\frac {1}{2} \left (-\int _1^x\frac {(K[1]-2) K[1]+2}{K[1]^2}dK[1]+x-\frac {2}{x}\right )\right ) \left (\int _1^x-\frac {\exp \left (\frac {1}{2} \left (K[3]+\int _1^{K[3]}\frac {(K[1]-2) K[1]+2}{K[1]^2}dK[1]-\frac {2}{K[3]}\right )\right ) \sec (K[3]) \int _1^{K[3]}e^{\frac {2}{K[2]}-K[2]} K[2]^2dK[2]}{K[3]}dK[3]+\int _1^xe^{\frac {2}{K[2]}-K[2]} K[2]^2dK[2] \left (\int _1^x\frac {\exp \left (\frac {1}{2} \left (K[4]+\int _1^{K[4]}\frac {(K[1]-2) K[1]+2}{K[1]^2}dK[1]-\frac {2}{K[4]}\right )\right ) \sec (K[4])}{K[4]}dK[4]+c_2\right )+c_1\right )}{x} \end{align*}

✗ Sympy

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

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