86.11.4 problem 4

Internal problem ID [23198]
Book : An introduction to Differential Equations. By Howard Frederick Cleaves. 1969. Oliver and Boyd publisher. ISBN 0050015044
Section : Chapter 9. The operational method. Exercise 9c at page 137
Problem number : 4
Date solved : Thursday, October 02, 2025 at 09:24:16 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+3 y^{\prime }+2 y&=x \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.014 (sec). Leaf size: 21

ode:=diff(diff(y(x),x),x)+3*diff(y(x),x)+2*y(x) = x; 
ic:=[y(0) = 1, D(y)(0) = 1]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = -\frac {3}{4}+\frac {x}{2}-\frac {9 \,{\mathrm e}^{-2 x}}{4}+4 \,{\mathrm e}^{-x} \]

✓ Mathematica. Time used: 0.01 (sec). Leaf size: 28

ode=D[y[x],{x,2}]+3*D[y[x],x]+2*y[x]==x; 
ic={y[0]==1,Derivative[1][y][0] ==1}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{4} \left (2 x-9 e^{-2 x}+16 e^{-x}-3\right ) \end{align*}

✓ Sympy. Time used: 0.124 (sec). Leaf size: 22

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x + 2*y(x) + 3*Derivative(y(x), x) + Derivative(y(x), (x, 2)),0) 
ics = {y(0): 1, Subs(Derivative(y(x), x), x, 0): 1} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \frac {x}{2} - \frac {3}{4} + 4 e^{- x} - \frac {9 e^{- 2 x}}{4} \]