72.7.19 problem 19

Internal problem ID [14681]
Book : DIFFERENTIAL EQUATIONS by Paul Blanchard, Robert L. Devaney, Glen R. Hall. 4th edition. Brooks/Cole. Boston, USA. 2012
Section : Chapter 1. First-Order Differential Equations. Exercises section 1.9 page 133
Problem number : 19
Date solved : Thursday, March 13, 2025 at 04:14:18 AM
CAS classification : [_linear]

\begin{align*} y^{\prime }&=a t y+4 \,{\mathrm e}^{-t^{2}} \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 45

ode:=diff(y(t),t) = a*t*y(t)+4*exp(-t^2); 
dsolve(ode,y(t), singsol=all);
 

\[ y = \frac {\left (4 \,\operatorname {erf}\left (\frac {\sqrt {2 a +4}\, t}{2}\right ) \sqrt {\pi }+c_{1} \sqrt {2 a +4}\right ) {\mathrm e}^{\frac {a \,t^{2}}{2}}}{\sqrt {2 a +4}} \]

✓ Mathematica. Time used: 0.114 (sec). Leaf size: 58

ode=D[y[t],t]==a*t*y[t]+4*Exp[-t^2]; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to \frac {e^{\frac {a t^2}{2}} \left (2 \sqrt {2 \pi } \text {erf}\left (\frac {\sqrt {a+2} t}{\sqrt {2}}\right )+\sqrt {a+2} c_1\right )}{\sqrt {a+2}} \]

✓ Sympy. Time used: 3.300 (sec). Leaf size: 78

from sympy import * 
t = symbols("t") 
a = symbols("a") 
y = Function("y") 
ode = Eq(-a*t*y(t) + Derivative(y(t), t) - 4*exp(-t**2),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \begin {cases} \frac {\tilde {\infty } C_{1}}{t^{2}} + \frac {\tilde {\infty } \operatorname {erf}{\left (t e^{0} \right )}}{t^{2}} & \text {for}\: a = 0 \\C_{1} e^{\frac {a t^{2}}{2}} + \frac {2 \sqrt {2} \sqrt {\pi } e^{\frac {a t^{2}}{2}} \operatorname {erf}{\left (\frac {\sqrt {2} t \sqrt {\operatorname {polar\_lift}{\left (a + 2 \right )}}}{2} \right )}}{\sqrt {\operatorname {polar\_lift}{\left (a + 2 \right )}}} & \text {otherwise} \end {cases} \]