60.9.25 problem 1880

Internal problem ID [11879]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 8, system of first order odes
Problem number : 1880
Date solved : Tuesday, January 28, 2025 at 06:23:57 PM
CAS classification : system_of_ODEs

\begin{align*} t^{2} \left (1-\sin \left (t \right )\right ) \left (\frac {d}{d t}x \left (t \right )\right )&=t \left (1-2 \sin \left (t \right )\right ) x \left (t \right )+t^{2} y \left (t \right )\\ t^{2} \left (1-\sin \left (t \right )\right ) \left (\frac {d}{d t}y \left (t \right )\right )&=\left (t \cos \left (t \right )-\sin \left (t \right )\right ) x \left (t \right )+t \left (1-t \cos \left (t \right )\right ) y \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.165 (sec). Leaf size: 22

dsolve([t^2*(1-sin(t))*diff(x(t),t)=t*(1-2*sin(t))*x(t)+t^2*y(t),t^2*(1-sin(t))*diff(y(t),t)=(t*cos(t)-sin(t))*x(t)+t*(1-t*cos(t))*y(t)],singsol=all)
 

\begin{align*} x \left (t \right ) &= t \left (c_{1} t +c_{2} \right ) \\ y \left (t \right ) &= c_{2} \sin \left (t \right )+c_{1} t \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.007 (sec). Leaf size: 27

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

\begin{align*} x(t)\to t (c_1 t+c_2) \\ y(t)\to c_1 t+c_2 \sin (t) \\ \end{align*}