89.23.30 problem 34

Internal problem ID [24834]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 10. Nonhomogeneous Equations: Operational methods. Miscellaneous Exercises at page 162
Problem number : 34
Date solved : Thursday, October 02, 2025 at 10:48:27 PM
CAS classification : [[_2nd_order, _missing_y]]

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

With initial conditions

\begin{align*} y \left (0\right )&=0 \\ y^{\prime }\left (0\right )&=1 \\ \end{align*}

✓ Maple. Time used: 0.008 (sec). Leaf size: 20

ode:=diff(diff(y(x),x),x)+2*diff(y(x),x) = 2*x; 
ic:=[y(0) = 0, D(y)(0) = 1]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = \frac {x^{2}}{2}-\frac {3 \,{\mathrm e}^{-2 x}}{4}-\frac {x}{2}+\frac {3}{4} \]

✓ Mathematica. Time used: 0.019 (sec). Leaf size: 26

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

\begin{align*} y(x)&\to \frac {1}{4} \left (2 x^2-2 x-3 e^{-2 x}+3\right ) \end{align*}

✓ Sympy. Time used: 0.120 (sec). Leaf size: 22

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-2*x + 2*Derivative(y(x), x) + Derivative(y(x), (x, 2)),0) 
ics = {y(0): 0, Subs(Derivative(y(x), x), x, 0): 1} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \frac {x^{2}}{2} - \frac {x}{2} + \frac {3}{4} - \frac {3 e^{- 2 x}}{4} \]