74.3.27 problem 22

Internal problem ID [15887]
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 : Tuesday, January 28, 2025 at 08:17:21 AM
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*}

✓ Solution by Maple

Time used: 0.497 (sec). Leaf size: 31

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

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

✓ Solution by Mathematica

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

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

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