76.2.15 problem 15

Internal problem ID [17280]
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 : 15
Date solved : Monday, March 31, 2025 at 03:48:35 PM
CAS classification : [_linear]

\begin{align*} t y^{\prime }+4 y&=t^{2}-t +1 \end{align*}

With initial conditions

\begin{align*} y \left (1\right )&={\frac {1}{4}} \end{align*}

✓ Maple. Time used: 0.023 (sec). Leaf size: 19

ode:=t*diff(y(t),t)+4*y(t) = t^2-t+1; 
ic:=y(1) = 1/4; 
dsolve([ode,ic],y(t), singsol=all);
 

\[ y = \frac {t^{2}}{6}-\frac {t}{5}+\frac {1}{4}+\frac {1}{30 t^{4}} \]

✓ Mathematica. Time used: 0.049 (sec). Leaf size: 24

ode=t*D[y[t],t]+4*y[t]==t^2-t+1; 
ic={y[1]==1/4}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to \frac {1}{60} \left (\frac {2}{t^4}+10 t^2-12 t+15\right ) \]

✓ Sympy. Time used: 0.224 (sec). Leaf size: 20

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-t**2 + t*Derivative(y(t), t) + t + 4*y(t) - 1,0) 
ics = {y(1): 1/4} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \frac {t^{2}}{6} - \frac {t}{5} + \frac {1}{4} + \frac {1}{30 t^{4}} \]