89.16.28 problem 28

Internal problem ID [24678]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 9. Nonhomogeneous Equations: Undetermined coefficients. Exercises at page 140
Problem number : 28
Date solved : Thursday, October 02, 2025 at 10:46:58 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }-4 y&=2-8 x \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=0 \\ y^{\prime }\left (0\right )&=5 \\ \end{align*}

✓ Maple. Time used: 0.012 (sec). Leaf size: 19

ode:=diff(diff(y(x),x),x)-4*y(x) = 2-8*x; 
ic:=[y(0) = 0, D(y)(0) = 5]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = {\mathrm e}^{2 x}-\frac {{\mathrm e}^{-2 x}}{2}+2 x -\frac {1}{2} \]

✓ Mathematica. Time used: 0.008 (sec). Leaf size: 26

ode=D[y[x],{x,2}]-4*y[x]== 2-8*x; 
ic={y[0]==0,Derivative[1][y][0] ==5}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to 2 x-\frac {e^{-2 x}}{2}+e^{2 x}-\frac {1}{2} \end{align*}

✓ Sympy. Time used: 0.076 (sec). Leaf size: 22

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(8*x - 4*y(x) + Derivative(y(x), (x, 2)) - 2,0) 
ics = {y(0): 0, Subs(Derivative(y(x), x), x, 0): 5} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = 2 x + e^{2 x} - \frac {1}{2} - \frac {e^{- 2 x}}{2} \]