57.5.19 problem 4

Internal problem ID [14374]
Book : A First Course in Differential Equations by J. David Logan. Third Edition. Springer-Verlag, NY. 2015.
Section : Chapter 1, First order differential equations. Section 1.4.1. Integrating factors. Exercises page 41
Problem number : 4
Date solved : Thursday, October 02, 2025 at 09:34:16 AM
CAS classification : [[_linear, `class A`]]

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

✓ Maple. Time used: 0.001 (sec). Leaf size: 60

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

\[ y = \frac {2 \,{\mathrm e}^{-a t} \left (-a \right )^{{3}/{2}} c_1 -2 \sqrt {-a}\, \sqrt {1+t}+{\mathrm e}^{-\left (1+t \right ) a} \operatorname {erf}\left (\sqrt {-a}\, \sqrt {1+t}\right ) \sqrt {\pi }}{2 \left (-a \right )^{{3}/{2}}} \]

✓ Mathematica. Time used: 0.153 (sec). Leaf size: 49

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

\begin{align*} y(t)&\to e^{-a t} \left (\frac {a e^{-a} (t+1)^{5/2} \Gamma \left (\frac {3}{2},-a (t+1)\right )}{(-a (t+1))^{5/2}}+c_1\right ) \end{align*}

✓ Sympy. Time used: 3.797 (sec). Leaf size: 19

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

\[ \int \left (a y{\left (t \right )} - \sqrt {t + 1}\right ) e^{a t}\, dt = C_{1} \]