14.1.5 problem 5

Internal problem ID [2476]
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 : 5
Date solved : Sunday, March 30, 2025 at 12:02:35 AM
CAS classification : [_linear]

\begin{align*} y^{\prime }+t^{2} y&=1 \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 57

ode:=diff(y(t),t)+t^2*y(t) = 1; 
dsolve(ode,y(t), singsol=all);
 

\[ y = -\frac {\left (3^{{1}/{3}} t \Gamma \left (\frac {1}{3}, -\frac {t^{3}}{3}\right ) \Gamma \left (\frac {2}{3}\right )-\frac {2 \,3^{{5}/{6}} t \pi }{3}-3 c_1 \Gamma \left (\frac {2}{3}\right ) \left (-t^{3}\right )^{{1}/{3}}\right ) {\mathrm e}^{-\frac {t^{3}}{3}}}{3 \left (-t^{3}\right )^{{1}/{3}} \Gamma \left (\frac {2}{3}\right )} \]

✓ Mathematica. Time used: 0.04 (sec). Leaf size: 52

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

\[ y(t)\to \frac {1}{3} e^{-\frac {t^3}{3}} \left (\frac {\sqrt [3]{3} \left (-t^3\right )^{2/3} \Gamma \left (\frac {1}{3},-\frac {t^3}{3}\right )}{t^2}+3 c_1\right ) \]

✓ Sympy. Time used: 0.793 (sec). Leaf size: 37

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

\[ y{\left (t \right )} = \left (C_{1} + \frac {\sqrt [3]{3} \gamma \left (\frac {1}{3}, \frac {t^{3} e^{i \pi }}{3}\right )}{3}\right ) e^{- \frac {t^{3}}{3} - \frac {i \pi }{3}} \]