8.1.12 problem 12

Internal problem ID [2483]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 1. First order differential equations. Section 1.2. Linear equations. Excercises page 9
Problem number : 12
Date solved : Tuesday, September 30, 2025 at 05:36:44 AM
CAS classification : [_linear]

\begin{align*} y^{\prime }+t y&=1+t \end{align*}

With initial conditions

\begin{align*} y \left (\frac {3}{2}\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.114 (sec). Leaf size: 46

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

\[ y = 1-{\mathrm e}^{\frac {9}{8}-\frac {t^{2}}{2}}-\frac {\sqrt {2}\, \sqrt {\pi }\, \left (i \operatorname {erf}\left (\frac {i \sqrt {2}\, t}{2}\right )+\operatorname {erfi}\left (\frac {3 \sqrt {2}}{4}\right )\right ) {\mathrm e}^{-\frac {t^{2}}{2}}}{2} \]

✓ Mathematica. Time used: 0.022 (sec). Leaf size: 72

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

\begin{align*} y(t)&\to \frac {1}{2} e^{-\frac {t^2}{2}} \left (\sqrt {2 \pi } \text {erfi}\left (\frac {t}{\sqrt {2}}\right )-\sqrt {2 \pi } \text {erfi}\left (\frac {3}{2 \sqrt {2}}\right )+2 e^{\frac {t^2}{2}}-2 e^{9/8}\right ) \end{align*}

✓ Sympy. Time used: 0.280 (sec). Leaf size: 68

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

\[ y{\left (t \right )} = 1 + \frac {\sqrt {2} \sqrt {\pi } e^{- \frac {t^{2}}{2}} \operatorname {erfi}{\left (\frac {\sqrt {2} t}{2} \right )}}{2} + \left (- e^{\frac {9}{8}} - \frac {\sqrt {2} \sqrt {\pi } \operatorname {erfi}{\left (\frac {3 \sqrt {2}}{4} \right )}}{2}\right ) e^{- \frac {t^{2}}{2}} \]