Internal
problem
ID
[10909]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
9,
system
of
higher
order
odes
Problem
number
:
1938
Date
solved
:
Friday, October 11, 2024 at 12:20:33 PM
CAS
classification
:
system_of_ODEs
\begin{align*} \left (x \left (t \right )-y \left (t \right )\right ) \left (x \left (t \right )-z \left (t \right )\right ) \left (\frac {d}{d t}x \left (t \right )\right )&=f \left (t \right )\\ \left (y \left (t \right )-x \left (t \right )\right ) \left (y \left (t \right )-z \left (t \right )\right ) \left (\frac {d}{d t}y \left (t \right )\right )&=f \left (t \right )\\ \left (z \left (t \right )-x \left (t \right )\right ) \left (z \left (t \right )-y \left (t \right )\right ) \left (\frac {d}{d t}z \left (t \right )\right )&=f \left (t \right ) \end{align*}
10.25.2 Maple dsolve solution
Solving time : 1.796 (sec)
Leaf size : 1110
dsolve([(x(t)-y(t))*(-z(t)+x(t))*diff(x(t),t) = f(t), (y(t)-x(t))*(y(t)-z(t))*diff(y(t),t) = f(t), (z(t)-x(t))*(z(t)-y(t))*diff(z(t),t) = f(t)]
,{op([x(t), y(t), z(t)])})
\begin{align*}
\text {Expression too large to display} \\
\left \{y \left (t \right ) &= \frac {4 \left (\frac {d}{d t}x \left (t \right )\right )^{3} x \left (t \right )+\left (\frac {d^{2}}{d t^{2}}x \left (t \right )\right ) f \left (t \right )-\left (\frac {d}{d t}f \left (t \right )\right ) \left (\frac {d}{d t}x \left (t \right )\right )-\sqrt {-16 \left (\frac {d}{d t}x \left (t \right )\right )^{5} f \left (t \right )+\left (\frac {d}{d t}x \left (t \right )\right )^{2} \left (\frac {d}{d t}f \left (t \right )\right )^{2}-2 \left (\frac {d}{d t}x \left (t \right )\right ) \left (\frac {d^{2}}{d t^{2}}x \left (t \right )\right ) f \left (t \right ) \left (\frac {d}{d t}f \left (t \right )\right )+\left (\frac {d^{2}}{d t^{2}}x \left (t \right )\right )^{2} f \left (t \right )^{2}}}{4 \left (\frac {d}{d t}x \left (t \right )\right )^{3}}, y \left (t \right ) &= \frac {4 \left (\frac {d}{d t}x \left (t \right )\right )^{3} x \left (t \right )+\left (\frac {d^{2}}{d t^{2}}x \left (t \right )\right ) f \left (t \right )-\left (\frac {d}{d t}f \left (t \right )\right ) \left (\frac {d}{d t}x \left (t \right )\right )+\sqrt {-16 \left (\frac {d}{d t}x \left (t \right )\right )^{5} f \left (t \right )+\left (\frac {d}{d t}x \left (t \right )\right )^{2} \left (\frac {d}{d t}f \left (t \right )\right )^{2}-2 \left (\frac {d}{d t}x \left (t \right )\right ) \left (\frac {d^{2}}{d t^{2}}x \left (t \right )\right ) f \left (t \right ) \left (\frac {d}{d t}f \left (t \right )\right )+\left (\frac {d^{2}}{d t^{2}}x \left (t \right )\right )^{2} f \left (t \right )^{2}}}{4 \left (\frac {d}{d t}x \left (t \right )\right )^{3}}\right \} \\
\left \{z \left (t \right ) &= \frac {-\left (\frac {d}{d t}x \left (t \right )\right ) y \left (t \right ) x \left (t \right )+\left (\frac {d}{d t}x \left (t \right )\right ) x \left (t \right )^{2}-f \left (t \right )}{-y \left (t \right ) \left (\frac {d}{d t}x \left (t \right )\right )+x \left (t \right ) \left (\frac {d}{d t}x \left (t \right )\right )}\right \} \\
\end{align*}
10.25.3 Mathematica DSolve solution
Solving time : 0.179 (sec)
Leaf size : 1557
DSolve[{{(x[t]-y[t])*(x[t]-z[t])*D[x[t],t]==f[t],(y[t]-x[t])*(y[t]-z[t])*D[y[t],t]==f[t],(z[t]-x[t])*(z[t]-y[t])*D[z[t],t]==f[t]},{}},
{x[t],y[t],z[t]},t,IncludeSingularSolutions->True]
\begin{align*}
x(t)\to \frac {1}{6} \left (2^{2/3} \sqrt [3]{27 \int _1^tf(K[1])dK[1]+\sqrt {\left (27 \int _1^tf(K[1])dK[1]+2 c_1{}^3-9 c_2 c_1+27 c_3\right ){}^2-4 \left (c_1{}^2-3 c_2\right ){}^3}+2 c_1{}^3-9 c_2 c_1+27 c_3}+\frac {2 \sqrt [3]{2} \left (c_1{}^2-3 c_2\right )}{\sqrt [3]{27 \int _1^tf(K[1])dK[1]+\sqrt {\left (27 \int _1^tf(K[1])dK[1]+2 c_1{}^3-9 c_2 c_1+27 c_3\right ){}^2-4 \left (c_1{}^2-3 c_2\right ){}^3}+2 c_1{}^3-9 c_2 c_1+27 c_3}}+2 c_1\right ) \\
y(t)\to -\frac {\sqrt {-\frac {-8 c_1{}^2 \left (27 \int _1^tf(K[1])dK[1]+\sqrt {\left (27 \int _1^tf(K[1])dK[1]+2 c_1{}^3-9 c_2 c_1+27 c_3\right ){}^2-4 \left (c_1{}^2-3 c_2\right ){}^3}+2 c_1{}^3-9 c_2 c_1+27 c_3\right ){}^{2/3}+2 \sqrt [3]{2} \left (27 \int _1^tf(K[1])dK[1]+\sqrt {\left (27 \int _1^tf(K[1])dK[1]+2 c_1{}^3-9 c_2 c_1+27 c_3\right ){}^2-4 \left (c_1{}^2-3 c_2\right ){}^3}+2 c_1{}^3-9 c_2 c_1+27 c_3\right ){}^{4/3}+24 c_2 \left (27 \int _1^tf(K[1])dK[1]+\sqrt {\left (27 \int _1^tf(K[1])dK[1]+2 c_1{}^3-9 c_2 c_1+27 c_3\right ){}^2-4 \left (c_1{}^2-3 c_2\right ){}^3}+2 c_1{}^3-9 c_2 c_1+27 c_3\right ){}^{2/3}+4\ 2^{2/3} c_1{}^4-24\ 2^{2/3} c_2 c_1{}^2+36\ 2^{2/3} c_2{}^2}{\left (27 \int _1^tf(K[1])dK[1]+\sqrt {\left (27 \int _1^tf(K[1])dK[1]+2 c_1{}^3-9 c_2 c_1+27 c_3\right ){}^2-4 \left (c_1{}^2-3 c_2\right ){}^3}+2 c_1{}^3-9 c_2 c_1+27 c_3\right ){}^{2/3}}}}{4 \sqrt {3}}-\frac {\sqrt [3]{27 \int _1^tf(K[1])dK[1]+\sqrt {\left (27 \int _1^tf(K[1])dK[1]+2 c_1{}^3-9 c_2 c_1+27 c_3\right ){}^2-4 \left (c_1{}^2-3 c_2\right ){}^3}+2 c_1{}^3-9 c_2 c_1+27 c_3}}{6 \sqrt [3]{2}}-\frac {c_1{}^2-3 c_2}{3\ 2^{2/3} \sqrt [3]{27 \int _1^tf(K[1])dK[1]+\sqrt {\left (27 \int _1^tf(K[1])dK[1]+2 c_1{}^3-9 c_2 c_1+27 c_3\right ){}^2-4 \left (c_1{}^2-3 c_2\right ){}^3}+2 c_1{}^3-9 c_2 c_1+27 c_3}}+\frac {c_1}{3} \\
z(t)\to \frac {4 c_1 \sqrt [3]{27 \int _1^tf(K[1])dK[1]+\sqrt {\left (27 \int _1^tf(K[1])dK[1]+2 c_1{}^3-9 c_2 c_1+27 c_3\right ){}^2-4 \left (c_1{}^2-3 c_2\right ){}^3}+2 c_1{}^3-9 c_2 c_1+27 c_3}-2^{2/3} \left (27 \int _1^tf(K[1])dK[1]+\sqrt {\left (27 \int _1^tf(K[1])dK[1]+2 c_1{}^3-9 c_2 c_1+27 c_3\right ){}^2-4 \left (c_1{}^2-3 c_2\right ){}^3}+2 c_1{}^3-9 c_2 c_1+27 c_3\right ){}^{2/3}+\sqrt {3} \sqrt [3]{27 \int _1^tf(K[1])dK[1]+\sqrt {\left (27 \int _1^tf(K[1])dK[1]+2 c_1{}^3-9 c_2 c_1+27 c_3\right ){}^2-4 \left (c_1{}^2-3 c_2\right ){}^3}+2 c_1{}^3-9 c_2 c_1+27 c_3} \sqrt {-\frac {-8 c_1{}^2 \left (27 \int _1^tf(K[1])dK[1]+\sqrt {\left (27 \int _1^tf(K[1])dK[1]+2 c_1{}^3-9 c_2 c_1+27 c_3\right ){}^2-4 \left (c_1{}^2-3 c_2\right ){}^3}+2 c_1{}^3-9 c_2 c_1+27 c_3\right ){}^{2/3}+2 \sqrt [3]{2} \left (27 \int _1^tf(K[1])dK[1]+\sqrt {\left (27 \int _1^tf(K[1])dK[1]+2 c_1{}^3-9 c_2 c_1+27 c_3\right ){}^2-4 \left (c_1{}^2-3 c_2\right ){}^3}+2 c_1{}^3-9 c_2 c_1+27 c_3\right ){}^{4/3}+24 c_2 \left (27 \int _1^tf(K[1])dK[1]+\sqrt {\left (27 \int _1^tf(K[1])dK[1]+2 c_1{}^3-9 c_2 c_1+27 c_3\right ){}^2-4 \left (c_1{}^2-3 c_2\right ){}^3}+2 c_1{}^3-9 c_2 c_1+27 c_3\right ){}^{2/3}+4\ 2^{2/3} c_1{}^4-24\ 2^{2/3} c_2 c_1{}^2+36\ 2^{2/3} c_2{}^2}{\left (27 \int _1^tf(K[1])dK[1]+\sqrt {\left (27 \int _1^tf(K[1])dK[1]+2 c_1{}^3-9 c_2 c_1+27 c_3\right ){}^2-4 \left (c_1{}^2-3 c_2\right ){}^3}+2 c_1{}^3-9 c_2 c_1+27 c_3\right ){}^{2/3}}}-2 \sqrt [3]{2} c_1{}^2+6 \sqrt [3]{2} c_2}{12 \sqrt [3]{27 \int _1^tf(K[1])dK[1]+\sqrt {\left (27 \int _1^tf(K[1])dK[1]+2 c_1{}^3-9 c_2 c_1+27 c_3\right ){}^2-4 \left (c_1{}^2-3 c_2\right ){}^3}+2 c_1{}^3-9 c_2 c_1+27 c_3}} \\
\end{align*}