54.3.77 problem 1091

Internal problem ID [12372]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1091
Date solved : Wednesday, October 01, 2025 at 01:44:03 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 35

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

\[ y = \frac {\sin \left (x \right ) \operatorname {Ci}\left (2 x \right )}{2}-\frac {\operatorname {Si}\left (2 x \right ) \cos \left (x \right )}{2}+\frac {\left (2 c_2 +\ln \left (x \right )\right ) \sin \left (x \right )}{2}+\cos \left (x \right ) c_1 \]

✓ Mathematica. Time used: 0.039 (sec). Leaf size: 56

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

\begin{align*} y(x)&\to \cos (x) \int _1^x-\frac {\cos (K[1]) \sin (K[1])}{K[1]}dK[1]+\frac {1}{2} \operatorname {CosIntegral}(2 x) \sin (x)+\frac {1}{2} \log (x) \sin (x)+c_1 \cos (x)+c_2 \sin (x) \end{align*}

✗ Sympy

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

NotImplementedError : solve: Cannot solve x*(y(x) + Derivative(y(x), (x, 2))) - cos(x)