74.3.27 problem 22

Internal problem ID [15816]
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 : 22
Date solved : Monday, March 31, 2025 at 01:56:32 PM
CAS classification : [_linear]

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

With initial conditions

\begin{align*} y \left (3\right )&=-1 \end{align*}

✓ Maple. Time used: 0.079 (sec). Leaf size: 31

ode:=diff(y(t),t)+y(t)/(-t^2+4)^(1/2) = t; 
ic:=y(3) = -1; 
dsolve([ode,ic],y(t), singsol=all);
 

\[ y = \left (\int _{3}^{t}\textit {\_z1} \,{\mathrm e}^{\arcsin \left (\frac {\textit {\_z1}}{2}\right )}d \textit {\_z1} -{\mathrm e}^{\arcsin \left (\frac {3}{2}\right )}\right ) {\mathrm e}^{-\arcsin \left (\frac {t}{2}\right )} \]

✓ Mathematica. Time used: 0.085 (sec). Leaf size: 82

ode=D[y[t],t]+y[t]/Sqrt[4-t^2]==t; 
ic={y[3]==-1}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to e^{-\arctan \left (\frac {t}{\sqrt {4-t^2}}\right )-i \text {arctanh}\left (\frac {3}{\sqrt {5}}\right )} \left (-1+e^{i \text {arctanh}\left (\frac {3}{\sqrt {5}}\right )} \int _3^te^{\arctan \left (\frac {K[1]}{\sqrt {4-K[1]^2}}\right )} K[1]dK[1]\right ) \]

✓ Sympy. Time used: 0.531 (sec). Leaf size: 60

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(3): -1} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \frac {2 t^{2}}{5} + \frac {t \sqrt {4 - t^{2}}}{5} - \frac {4}{5} + \left (- \frac {3 \sqrt {5} i e^{\operatorname {asin}{\left (\frac {3}{2} \right )}}}{5} - \frac {19 e^{\operatorname {asin}{\left (\frac {3}{2} \right )}}}{5}\right ) e^{- \operatorname {asin}{\left (\frac {t}{2} \right )}} \]