85.5.1 problem 3

Internal problem ID [22447]
Book : Applied Differential Equations. By Murray R. Spiegel. 3rd edition. 1980. Pearson. ISBN 978-0130400970
Section : Chapter 1. Differential equations in general. Section 1.3. B Exercises at page 22
Problem number : 3
Date solved : Thursday, October 02, 2025 at 08:39:39 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

With initial conditions

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

✓ Maple. Time used: 0.037 (sec). Leaf size: 46

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

\[ y = \frac {\sqrt {\pi }\, \left (\left (i \cos \left (x \right )+\sin \left (x \right )\right ) \operatorname {erf}\left (x -\frac {i}{2}\right )+\left (-i \cos \left (x \right )+\sin \left (x \right )\right ) \operatorname {erf}\left (x +\frac {i}{2}\right )-2 \,\operatorname {erfi}\left (\frac {1}{2}\right ) \cos \left (x \right )\right ) {\mathrm e}^{-\frac {1}{4}}}{4} \]

✓ Mathematica. Time used: 0.048 (sec). Leaf size: 76

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

\begin{align*} y(x)&\to \frac {\sqrt {\pi } \left (\text {erf}\left (\frac {1}{2} (2 x+i)\right ) \sin (x)+\text {erfi}\left (\frac {1}{2}-i x\right ) \cos (x)-2 \text {erfi}\left (\frac {1}{2}\right ) \cos (x)+\text {erfi}\left (\frac {1}{2}+i x\right ) (\cos (x)-i \sin (x))\right )}{4 \sqrt [4]{e}} \end{align*}

✓ Sympy. Time used: 2.200 (sec). Leaf size: 49

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

\[ y{\left (x \right )} = \left (- \int e^{- x^{2}} \sin {\left (x \right )}\, dx + \int \limits ^{0} e^{- x^{2}} \sin {\left (x \right )}\, dx\right ) \cos {\left (x \right )} + \left (\int e^{- x^{2}} \cos {\left (x \right )}\, dx - \int \limits ^{0} e^{- x^{2}} \cos {\left (x \right )}\, dx\right ) \sin {\left (x \right )} \]