74.12.59 problem 65

Internal problem ID [16366]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 4. Higher Order Equations. Exercises 4.4, page 163
Problem number : 65
Date solved : Tuesday, January 28, 2025 at 09:07:07 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

With initial conditions

\begin{align*} y \left (\frac {\pi }{4}\right )&=0\\ y^{\prime }\left (\frac {\pi }{4}\right )&=0 \end{align*}

✓ Solution by Maple

Time used: 1.381 (sec). Leaf size: 29

dsolve([(sin(t)-t*cos(t))*diff(y(t),t$2)-t*sin(t)*diff(y(t),t)+sin(t)*y(t)=t,y(1/4*Pi) = 0, D(y)(1/4*Pi) = 0],y(t), singsol=all)
 

\[ y = \frac {-4 \sqrt {2}\, t +\left (\pi -4\right ) \cos \left (t \right )+\left (\pi +4\right ) \sin \left (t \right )}{\pi -4} \]

✓ Solution by Mathematica

Time used: 0.733 (sec). Leaf size: 109

DSolve[{(Sin[t]-t*Cos[t])*D[y[t],{t,2}]-t*Sin[t]*D[y[t],t]+Sin[t]*y[t]==t,{y[Pi/4]==0,Derivative[1][y][Pi/4]==0}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to -\sin (t) \int _1^{\frac {\pi }{4}}-\frac {K[1]^2}{(\sin (K[1])-\cos (K[1]) K[1])^2}dK[1]+\sin (t) \int _1^t-\frac {K[1]^2}{(\sin (K[1])-\cos (K[1]) K[1])^2}dK[1]+\frac {t \left (4 \sqrt {2} \sin (t)-4 \sqrt {2} t \cos (t)+\pi -4\right )}{(\pi -4) (t \cos (t)-\sin (t))} \]