60.9.42 problem 1897

Internal problem ID [11821]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 8, system of first order odes
Problem number : 1897
Date solved : Sunday, March 30, 2025 at 09:16:00 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.536 (sec). Leaf size: 113

ode:=[diff(diff(x(t),t),t)+diff(diff(y(t),t),t)+diff(y(t),t) = sinh(2*t), 2*diff(diff(x(t),t),t)+diff(diff(y(t),t),t) = 2*t]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= \frac {t^{2}}{4}+\frac {t^{3}}{6}+\frac {t \sinh \left (2 t \right )}{4}-\frac {3 \cosh \left (2 t \right )}{8}-\frac {t \cosh \left (2 t \right )}{4}+\frac {\sinh \left (2 t \right )}{4}+\frac {\cosh \left (2 t \right ) c_2}{4}-\frac {c_2 \sinh \left (2 t \right )}{4}+c_3 t +c_4 \\ y \left (t \right ) &= -\frac {t^{2}}{2}+\frac {t}{2}-\frac {\sinh \left (2 t \right )}{2}+\frac {t \cosh \left (2 t \right )}{2}+\frac {3 \cosh \left (2 t \right )}{4}-\frac {t \sinh \left (2 t \right )}{2}+\frac {c_2 \sinh \left (2 t \right )}{2}-\frac {\cosh \left (2 t \right ) c_2}{2}+c_3 +c_1 \\ \end{align*}

✓ Mathematica. Time used: 1.442 (sec). Leaf size: 118

ode={D[x[t],{t,2}]+D[y[t],{t,2}]+D[y[t],t]==Sinh[2*t],2*D[x[t],{t,2}]+D[y[t],{t,2}]==2*t}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)\to \frac {1}{48} \left (2 \left (4 t^3+6 t^2+6 (-1+4 c_2+2 c_4) t+3+24 c_1-6 c_4\right )-3 e^{2 t}-6 e^{-2 t} (2 t+1-2 c_4)\right ) \\ y(t)\to \frac {1}{8} e^{-2 t} \left (e^{2 t} \left (-4 t^2+4 t-2+8 c_3+4 c_4\right )+4 t+e^{4 t}+2-4 c_4\right ) \\ \end{align*}

✓ Sympy. Time used: 0.321 (sec). Leaf size: 104

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-sinh(2*t) + Derivative(x(t), (t, 2)) + Derivative(y(t), t) + Derivative(y(t), (t, 2)),0),Eq(-2*t + 2*Derivative(x(t), (t, 2)) + Derivative(y(t), (t, 2)),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = C_{1} + \frac {C_{3} e^{- 2 t}}{4} + \frac {t^{3}}{6} + \frac {t^{2}}{4} + t \left (C_{2} - \frac {1}{4}\right ) + \frac {t \sinh {\left (2 t \right )}}{4} - \frac {t \cosh {\left (2 t \right )}}{4} - \frac {\cosh {\left (2 t \right )}}{8} + \frac {1}{8}, \ y{\left (t \right )} = - \frac {C_{3} e^{- 2 t}}{2} + C_{4} - \frac {t^{2}}{2} - \frac {t \sinh {\left (2 t \right )}}{2} + \frac {t \cosh {\left (2 t \right )}}{2} + \frac {t}{2} + \frac {\cosh {\left (2 t \right )}}{4} - \frac {1}{4}\right ] \]