9.45 problem 1900

Internal problem ID [9479]

Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 8, system of first order odes
Problem number: 1900.
ODE order: 1.
ODE degree: 1.

Solve \begin {align*} x^{\prime }\relax (t )&=4 x \relax (t )\\ y^{\prime }\relax (t )&=x \relax (t )-2 y \relax (t )\\ z^{\prime }\relax (t )&=x \relax (t )-4 y \relax (t )+z \relax (t ) \end {align*}

Solution by Maple

Time used: 0.103 (sec). Leaf size: 50

dsolve({diff(x(t),t)=4*x(t),diff(y(t),t)=x(t)-2*y(t),diff(z(t),t)=x(t)-4*y(t)+z(t)},{x(t), y(t), z(t)}, singsol=all)
 

\[ x \relax (t ) = c_{3} {\mathrm e}^{4 t} \] \[ y \relax (t ) = \frac {c_{3} {\mathrm e}^{4 t}}{6}+c_{2} {\mathrm e}^{-2 t} \] \[ z \relax (t ) = \frac {c_{3} {\mathrm e}^{4 t}}{9}+c_{1} {\mathrm e}^{t}+\frac {4 c_{2} {\mathrm e}^{-2 t}}{3} \]

Solution by Mathematica

Time used: 0.006 (sec). Leaf size: 86

DSolve[{x'[t]==4*x[t],y'[t]==x[t]-2*y[t],z'[t]==x[t]-4*y[t]+z[t]},{x[t],y[t],z[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to c_1 e^{4 t} \\ y(t)\to \frac {1}{6} e^{-2 t} \left (c_1 \left (e^{6 t}-1\right )+6 c_2\right ) \\ z(t)\to \frac {1}{9} e^{-2 t} \left (c_1 e^{6 t}+(c_1-12 c_2+9 c_3) e^{3 t}-2 c_1+12 c_2\right ) \\ \end{align*}