74.12.59 problem 65

Internal problem ID [16295]
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 : Monday, March 31, 2025 at 02:49:50 PM
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 }+\sin \left (t \right ) y&=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*}

✓ Maple. Time used: 0.092 (sec). Leaf size: 29

ode:=(sin(t)-t*cos(t))*diff(diff(y(t),t),t)-t*sin(t)*diff(y(t),t)+y(t)*sin(t) = t; 
ic:=y(1/4*Pi) = 0, D(y)(1/4*Pi) = 0; 
dsolve([ode,ic],y(t), singsol=all);
 

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

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

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

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

✗ Sympy

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

NotImplementedError : The given ODE Derivative(y(t), t) - (-t*Derivative(y(t), (t, 2))/tan(t) - t/sin(t) + y(t) + Derivative(y(t), (t, 2)))/t cannot be solved by the factorable group method