85.7.4 problem 1 (d)

Internal problem ID [22458]
Book : Applied Differential Equations. By Murray R. Spiegel. 3rd edition. 1980. Pearson. ISBN 978-0130400970
Section : Chapter 1. Differential equations in general. A Exercises at page 32
Problem number : 1 (d)
Date solved : Thursday, October 02, 2025 at 08:39:49 PM
CAS classification : [_linear]

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

With initial conditions

\begin{align*} y \left (0\right )&=2 \\ \end{align*}

✓ Maple. Time used: 0.022 (sec). Leaf size: 37

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

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

✓ Mathematica. Time used: 0.042 (sec). Leaf size: 46

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

\begin{align*} y(x)&\to -\sqrt {\frac {\pi }{2}} e^{-\frac {x^2}{2}} \text {erfi}\left (\frac {x}{\sqrt {2}}\right )+2 e^{-\frac {x^2}{2}}+x \end{align*}

✓ Sympy. Time used: 0.337 (sec). Leaf size: 41

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

\[ y{\left (x \right )} = x - \frac {\sqrt {2} \sqrt {\pi } e^{- \frac {x^{2}}{2}} \operatorname {erfi}{\left (\frac {\sqrt {2} x}{2} \right )}}{2} + 2 e^{- \frac {x^{2}}{2}} \]