23.3.153 problem 155

Internal problem ID [5867]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 3. THE DIFFERENTIAL EQUATION IS LINEAR AND OF SECOND ORDER, page 311
Problem number : 155
Date solved : Tuesday, September 30, 2025 at 02:05:10 PM
CAS classification : [[_2nd_order, _exact, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.001 (sec). Leaf size: 40

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

\[ y = \left (c_2 +\frac {\int \left (4 c_1 -3 x^{2}+4 a x +2 x^{2} \ln \left (x \right )\right ) {\mathrm e}^{\cos \left (x \right )}d x}{4}\right ) {\mathrm e}^{-\cos \left (x \right )} \]

✓ Mathematica. Time used: 60.926 (sec). Leaf size: 56

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

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

✗ Sympy

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

NotImplementedError : The given ODE -(-a - x*log(x) + x - y(x)*cos(x) + Derivative(y(x), (x, 2)))/si