60.9.42 problem 1897

Internal problem ID [11896]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 8, system of first order odes
Problem number : 1897
Date solved : Tuesday, January 28, 2025 at 06:24:04 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*}

✓ Solution by Maple

Time used: 1.619 (sec). Leaf size: 113

dsolve([diff(x(t),t,t)+diff(y(t),t,t)+diff(y(t),t)=sinh(2*t),2*diff(x(t),t,t)+diff(y(t),t,t)=2*t],singsol=all)
 

\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}+t c_3 +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*}

✓ Solution by Mathematica

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

DSolve[{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},{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*}