34.10.11 problem 20

Internal problem ID [8020]
Book : Schaums Outline. Theory and problems of Differential Equations, 1st edition. Frank Ayres. McGraw Hill 1952
Section : Chapter 15. Linear equations with constant coefficients (Variation of parameters). Supplemetary problems. Page 98
Problem number : 20
Date solved : Tuesday, September 30, 2025 at 05:14:20 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 23

ode:=diff(diff(y(x),x),x)+y(x) = -2*sin(x)+4*x*cos(x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \left (x^{2}+c_2 -1\right ) \sin \left (x \right )+2 \left (x +\frac {c_1}{2}\right ) \cos \left (x \right ) \]

✓ Mathematica. Time used: 0.16 (sec). Leaf size: 70

ode=D[y[x],{x,2}]+y[x]==-2*Sin[x]+4*x*Cos[x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \sin (x) \int _1^x\left (4 \cos ^2(K[2]) K[2]-2 \cos (K[2]) \sin (K[2])\right )dK[2]+\cos (x) \int _1^x2 \sin (K[1]) (\sin (K[1])-2 \cos (K[1]) K[1])dK[1]+c_1 \cos (x)+c_2 \sin (x) \end{align*}

✓ Sympy. Time used: 0.069 (sec). Leaf size: 19

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

\[ y{\left (x \right )} = \left (C_{1} + 2 x\right ) \cos {\left (x \right )} + \left (C_{2} + x^{2}\right ) \sin {\left (x \right )} \]