74.6.27 problem 28

Internal problem ID [16050]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 2. First Order Equations. Exercises 2.4, page 57
Problem number : 28
Date solved : Tuesday, January 28, 2025 at 08:31:34 AM
CAS classification : [_exact, [_1st_order, `_with_symmetry_[F(x)*G(y),0]`], [_Abel, `2nd type`, `class B`]]

\begin{align*} \frac {2 t^{2} y \cos \left (t^{2}\right )-y \sin \left (t^{2}\right )}{t^{2}}+\frac {\left (2 t y+\sin \left (t^{2}\right )\right ) y^{\prime }}{t}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.128 (sec). Leaf size: 63

dsolve(1/t^2*(2*t^2*y(t)*cos(t^2)-y(t)*sin(t^2)  )+1/t*(2*t*y(t)+sin(t^2))*diff(y(t),t)=0,y(t), singsol=all)
 

\begin{align*} y &= \frac {-\sin \left (t^{2}\right )+\sqrt {\sin \left (t^{2}\right )^{2}-4 c_{1} t^{2}}}{2 t} \\ y &= \frac {-\sin \left (t^{2}\right )-\sqrt {\sin \left (t^{2}\right )^{2}-4 c_{1} t^{2}}}{2 t} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.625 (sec). Leaf size: 94

DSolve[1/t^2*(2*t^2*y[t]*Cos[t^2]-y[t]*Sin[t^2]  )+1/t*(2*t*y[t]+Sin[t^2])*D[y[t],t]==0,y[t],t,IncludeSingularSolutions -> True]
 

\begin{align*} y(t)\to -\frac {\sin \left (t^2\right )+\sqrt {\frac {1}{t^2}} t \sqrt {\sin ^2\left (t^2\right )+4 c_1 t^2}}{2 t} \\ y(t)\to \frac {-\sin \left (t^2\right )+\sqrt {\frac {1}{t^2}} t \sqrt {\sin ^2\left (t^2\right )+4 c_1 t^2}}{2 t} \\ y(t)\to 0 \\ \end{align*}