Internal problem ID [9454]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 8, system of first order odes
Problem number: 1875.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x^{\prime }\relax (t )&=-x \relax (t ) f \relax (t ) a -f \relax (t ) y \relax (t ) b +g \relax (t )\\ y^{\prime }\relax (t )&=-x \relax (t ) f \relax (t ) c -f \relax (t ) y \relax (t ) d +h \relax (t ) \end {align*}
✓ Solution by Maple
Time used: 2.218 (sec). Leaf size: 3360
dsolve({diff(x(t),t)+(a*x(t)+b*y(t))*f(t)=g(t),diff(y(t),t)+(c*x(t)+d*y(t))*f(t)=h(t)},{x(t), y(t)}, singsol=all)
\[ \text {Expression too large to display} \] \[ \text {Expression too large to display} \]
✓ Solution by Mathematica
Time used: 0.781 (sec). Leaf size: 2310
DSolve[{x'[t]+(a*x[t]+b*y[t])*f[t]==g[t],y'[t]+(c*x[t]+d*y[t])*f[t]==h[t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
\begin{align*} x(t)\to \frac {\left (-a+d+\sqrt {(a-d)^2+4 b c}\right ) e^{\int _1^t\frac {1}{2} \left (-a-d+\sqrt {(a-d)^2+4 b c}\right ) f(K[1])dK[1]} c_1+\left (a-d+\sqrt {(a-d)^2+4 b c}\right ) e^{\int _1^t-\frac {1}{2} \left (a+d+\sqrt {(a-d)^2+4 b c}\right ) f(K[2])dK[2]} c_2+\left (-a+d+\sqrt {(a-d)^2+4 b c}\right ) e^{\int _1^t\frac {1}{2} \left (-a-d+\sqrt {(a-d)^2+4 b c}\right ) f(K[1])dK[1]} \int _1^t\frac {\sqrt {(a-d)^2+4 b c} \left (2 c e^{\int _1^{K[5]}-\frac {1}{2} \left (a+d+\sqrt {(a-d)^2+4 b c}\right ) f(K[4])dK[4]} g(K[5])-\left (a-d+\sqrt {(a-d)^2+4 b c}\right ) e^{\int _1^{K[5]}-\frac {1}{2} \left (a+d+\sqrt {(a-d)^2+4 b c}\right ) f(K[2])dK[2]} h(K[5])\right )}{c \left (e^{\int _1^{K[5]}-\frac {1}{2} \left (a+d+\sqrt {(a-d)^2+4 b c}\right ) f(K[2])dK[2]+\int _1^{K[5]}\frac {1}{2} \left (-a-d+\sqrt {(a-d)^2+4 b c}\right ) f(K[3])dK[3]} (a-d)+\sqrt {(a-d)^2+4 b c} e^{\int _1^{K[5]}-\frac {1}{2} \left (a+d+\sqrt {(a-d)^2+4 b c}\right ) f(K[2])dK[2]+\int _1^{K[5]}\frac {1}{2} \left (-a-d+\sqrt {(a-d)^2+4 b c}\right ) f(K[3])dK[3]}+(d-a) e^{\int _1^{K[5]}\frac {1}{2} \left (-a-d+\sqrt {(a-d)^2+4 b c}\right ) f(K[1])dK[1]+\int _1^{K[5]}-\frac {1}{2} \left (a+d+\sqrt {(a-d)^2+4 b c}\right ) f(K[4])dK[4]}+\sqrt {(a-d)^2+4 b c} e^{\int _1^{K[5]}\frac {1}{2} \left (-a-d+\sqrt {(a-d)^2+4 b c}\right ) f(K[1])dK[1]+\int _1^{K[5]}-\frac {1}{2} \left (a+d+\sqrt {(a-d)^2+4 b c}\right ) f(K[4])dK[4]}\right )}dK[5]+\left (a-d+\sqrt {(a-d)^2+4 b c}\right ) e^{\int _1^t-\frac {1}{2} \left (a+d+\sqrt {(a-d)^2+4 b c}\right ) f(K[2])dK[2]} \int _1^t\frac {\sqrt {(a-d)^2+4 b c} \left (2 c e^{\int _1^{K[6]}\frac {1}{2} \left (-a-d+\sqrt {(a-d)^2+4 b c}\right ) f(K[3])dK[3]} g(K[6])+\left (-a+d+\sqrt {(a-d)^2+4 b c}\right ) e^{\int _1^{K[6]}\frac {1}{2} \left (-a-d+\sqrt {(a-d)^2+4 b c}\right ) f(K[1])dK[1]} h(K[6])\right )}{c \left (e^{\int _1^{K[6]}-\frac {1}{2} \left (a+d+\sqrt {(a-d)^2+4 b c}\right ) f(K[2])dK[2]+\int _1^{K[6]}\frac {1}{2} \left (-a-d+\sqrt {(a-d)^2+4 b c}\right ) f(K[3])dK[3]} (a-d)+\sqrt {(a-d)^2+4 b c} e^{\int _1^{K[6]}-\frac {1}{2} \left (a+d+\sqrt {(a-d)^2+4 b c}\right ) f(K[2])dK[2]+\int _1^{K[6]}\frac {1}{2} \left (-a-d+\sqrt {(a-d)^2+4 b c}\right ) f(K[3])dK[3]}+(d-a) e^{\int _1^{K[6]}\frac {1}{2} \left (-a-d+\sqrt {(a-d)^2+4 b c}\right ) f(K[1])dK[1]+\int _1^{K[6]}-\frac {1}{2} \left (a+d+\sqrt {(a-d)^2+4 b c}\right ) f(K[4])dK[4]}+\sqrt {(a-d)^2+4 b c} e^{\int _1^{K[6]}\frac {1}{2} \left (-a-d+\sqrt {(a-d)^2+4 b c}\right ) f(K[1])dK[1]+\int _1^{K[6]}-\frac {1}{2} \left (a+d+\sqrt {(a-d)^2+4 b c}\right ) f(K[4])dK[4]}\right )}dK[6]}{2 \sqrt {(a-d)^2+4 b c}} \\ y(t)\to \frac {c e^{\int _1^t-\frac {1}{2} \left (a+d+\sqrt {(a-d)^2+4 b c}\right ) f(K[4])dK[4]} \left (c_2+\int _1^t\frac {\sqrt {(a-d)^2+4 b c} \left (2 c e^{\int _1^{K[6]}\frac {1}{2} \left (-a-d+\sqrt {(a-d)^2+4 b c}\right ) f(K[3])dK[3]} g(K[6])+\left (-a+d+\sqrt {(a-d)^2+4 b c}\right ) e^{\int _1^{K[6]}\frac {1}{2} \left (-a-d+\sqrt {(a-d)^2+4 b c}\right ) f(K[1])dK[1]} h(K[6])\right )}{c \left (e^{\int _1^{K[6]}-\frac {1}{2} \left (a+d+\sqrt {(a-d)^2+4 b c}\right ) f(K[2])dK[2]+\int _1^{K[6]}\frac {1}{2} \left (-a-d+\sqrt {(a-d)^2+4 b c}\right ) f(K[3])dK[3]} (a-d)+\sqrt {(a-d)^2+4 b c} e^{\int _1^{K[6]}-\frac {1}{2} \left (a+d+\sqrt {(a-d)^2+4 b c}\right ) f(K[2])dK[2]+\int _1^{K[6]}\frac {1}{2} \left (-a-d+\sqrt {(a-d)^2+4 b c}\right ) f(K[3])dK[3]}+(d-a) e^{\int _1^{K[6]}\frac {1}{2} \left (-a-d+\sqrt {(a-d)^2+4 b c}\right ) f(K[1])dK[1]+\int _1^{K[6]}-\frac {1}{2} \left (a+d+\sqrt {(a-d)^2+4 b c}\right ) f(K[4])dK[4]}+\sqrt {(a-d)^2+4 b c} e^{\int _1^{K[6]}\frac {1}{2} \left (-a-d+\sqrt {(a-d)^2+4 b c}\right ) f(K[1])dK[1]+\int _1^{K[6]}-\frac {1}{2} \left (a+d+\sqrt {(a-d)^2+4 b c}\right ) f(K[4])dK[4]}\right )}dK[6]\right )-c e^{\int _1^t\frac {1}{2} \left (-a-d+\sqrt {(a-d)^2+4 b c}\right ) f(K[3])dK[3]} \left (c_1+\int _1^t\frac {\sqrt {(a-d)^2+4 b c} \left (2 c e^{\int _1^{K[5]}-\frac {1}{2} \left (a+d+\sqrt {(a-d)^2+4 b c}\right ) f(K[4])dK[4]} g(K[5])-\left (a-d+\sqrt {(a-d)^2+4 b c}\right ) e^{\int _1^{K[5]}-\frac {1}{2} \left (a+d+\sqrt {(a-d)^2+4 b c}\right ) f(K[2])dK[2]} h(K[5])\right )}{c \left (e^{\int _1^{K[5]}-\frac {1}{2} \left (a+d+\sqrt {(a-d)^2+4 b c}\right ) f(K[2])dK[2]+\int _1^{K[5]}\frac {1}{2} \left (-a-d+\sqrt {(a-d)^2+4 b c}\right ) f(K[3])dK[3]} (a-d)+\sqrt {(a-d)^2+4 b c} e^{\int _1^{K[5]}-\frac {1}{2} \left (a+d+\sqrt {(a-d)^2+4 b c}\right ) f(K[2])dK[2]+\int _1^{K[5]}\frac {1}{2} \left (-a-d+\sqrt {(a-d)^2+4 b c}\right ) f(K[3])dK[3]}+(d-a) e^{\int _1^{K[5]}\frac {1}{2} \left (-a-d+\sqrt {(a-d)^2+4 b c}\right ) f(K[1])dK[1]+\int _1^{K[5]}-\frac {1}{2} \left (a+d+\sqrt {(a-d)^2+4 b c}\right ) f(K[4])dK[4]}+\sqrt {(a-d)^2+4 b c} e^{\int _1^{K[5]}\frac {1}{2} \left (-a-d+\sqrt {(a-d)^2+4 b c}\right ) f(K[1])dK[1]+\int _1^{K[5]}-\frac {1}{2} \left (a+d+\sqrt {(a-d)^2+4 b c}\right ) f(K[4])dK[4]}\right )}dK[5]\right )}{\sqrt {(a-d)^2+4 b c}} \\ \end{align*}