68.3.26 problem 21

Internal problem ID [17173]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 2. First Order Equations. Exercises 2.1, page 32
Problem number : 21
Date solved : Thursday, October 02, 2025 at 01:49:10 PM
CAS classification : [_linear]

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

With initial conditions

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

✓ Maple. Time used: 0.041 (sec). Leaf size: 25

ode:=diff(y(t),t)+y(t)/(-t^2+4)^(1/2) = t; 
ic:=[y(0) = 0]; 
dsolve([ode,op(ic)],y(t), singsol=all);
 

\[ y = \int _{0}^{t}\textit {\_z1} \,{\mathrm e}^{\arcsin \left (\frac {\textit {\_z1}}{2}\right )}d \textit {\_z1} {\mathrm e}^{-\arcsin \left (\frac {t}{2}\right )} \]

✓ Mathematica. Time used: 0.066 (sec). Leaf size: 52

ode=D[y[t],t]+y[t]/Sqrt[4-t^2]==t; 
ic={y[0]==0}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to e^{-\arctan \left (\frac {t}{\sqrt {4-t^2}}\right )} \int _0^te^{\arctan \left (\frac {K[1]}{\sqrt {4-K[1]^2}}\right )} K[1]dK[1] \end{align*}

✓ Sympy. Time used: 0.295 (sec). Leaf size: 34

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

\[ y{\left (t \right )} = \frac {2 t^{2}}{5} + \frac {t \sqrt {4 - t^{2}}}{5} - \frac {4}{5} + \frac {4 e^{- \operatorname {asin}{\left (\frac {t}{2} \right )}}}{5} \]