34.10.6 problem 15

Internal problem ID [8015]
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 : 15
Date solved : Tuesday, September 30, 2025 at 05:14:16 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+2 y&=2+{\mathrm e}^{x} \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 26

ode:=diff(diff(y(x),x),x)+2*y(x) = 2+exp(x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \sin \left (\sqrt {2}\, x \right ) c_2 +\cos \left (\sqrt {2}\, x \right ) c_1 +1+\frac {{\mathrm e}^{x}}{3} \]

✓ Mathematica. Time used: 0.055 (sec). Leaf size: 103

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

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

✓ Sympy. Time used: 0.043 (sec). Leaf size: 29

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

\[ y{\left (x \right )} = C_{1} \sin {\left (\sqrt {2} x \right )} + C_{2} \cos {\left (\sqrt {2} x \right )} + \frac {e^{x}}{3} + 1 \]