76.2.13 problem 13

Internal problem ID [17278]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 2. First order differential equations. Section 2.2 (Linear equations: Method of integrating factors). Problems at page 54
Problem number : 13
Date solved : Monday, March 31, 2025 at 03:48:30 PM
CAS classification : [[_linear, `class A`]]

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

With initial conditions

\begin{align*} y \left (0\right )&=1 \end{align*}

✓ Maple. Time used: 0.026 (sec). Leaf size: 19

ode:=diff(y(t),t)-y(t) = 2*t*exp(2*t); 
ic:=y(0) = 1; 
dsolve([ode,ic],y(t), singsol=all);
 

\[ y = {\mathrm e}^{2 t} \left (2 t -2\right )+3 \,{\mathrm e}^{t} \]

✓ Mathematica. Time used: 0.053 (sec). Leaf size: 19

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

\[ y(t)\to e^t \left (2 e^t (t-1)+3\right ) \]

✓ Sympy. Time used: 0.179 (sec). Leaf size: 15

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

\[ y{\left (t \right )} = \left (2 \left (t - 1\right ) e^{t} + 3\right ) e^{t} \]