60.10.11 problem 1925
Internal
problem
ID
[11846]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
9,
system
of
higher
order
odes
Problem
number
:
1925
Date
solved
:
Friday, March 14, 2025 at 03:00:15 AM
CAS
classification
:
system_of_ODEs
\begin{align*} \left (\frac {d}{d t}x \left (t \right )\right )^{2}+t \left (\frac {d}{d t}x \left (t \right )\right )+a \left (\frac {d}{d t}y \left (t \right )\right )-x \left (t \right )&=0\\ \left (\frac {d}{d t}x \left (t \right )\right ) \left (\frac {d}{d t}y \left (t \right )\right )+t \left (\frac {d}{d t}y \left (t \right )\right )-y \left (t \right )&=0 \end{align*}
✓ Maple. Time used: 0.349 (sec). Leaf size: 220
ode:=[diff(x(t),t)^2+t*diff(x(t),t)+a*diff(y(t),t)-x(t) = 0, diff(x(t),t)*diff(y(t),t)+t*diff(y(t),t)-y(t) = 0];
dsolve(ode);
\begin{align*}
\left [\left \{x \left (t \right ) &= -\frac {t^{2}}{3}\right \}, \left \{y \left (t \right ) = -\frac {t^{3}}{27 a}\right \}\right ] \\
\left [\{x \left (t \right ) &= c_{1} t +c_{2}\}, \left \{y \left (t \right ) = \frac {-\left (\frac {d}{d t}x \left (t \right )\right )^{3}-2 \left (\frac {d}{d t}x \left (t \right )\right )^{2} t -t^{2} \left (\frac {d}{d t}x \left (t \right )\right )+x \left (t \right ) \left (\frac {d}{d t}x \left (t \right )\right )+x \left (t \right ) t}{a}\right \}\right ] \\
\left [\left \{x \left (t \right ) &= -\frac {5 t^{2}}{12}-\frac {\left (-t -\sqrt {3}\, c_{1} \right ) t}{6}+\frac {c_{1}^{2}}{4}, x \left (t \right ) = -\frac {5 t^{2}}{12}-\frac {\left (-t +\sqrt {3}\, c_{1} \right ) t}{6}+\frac {c_{1}^{2}}{4}, x \left (t \right ) = -\frac {5 t^{2}}{12}+\frac {\left (t -\sqrt {3}\, c_{1} \right ) t}{6}+\frac {c_{1}^{2}}{4}, x \left (t \right ) = -\frac {5 t^{2}}{12}+\frac {\left (t +\sqrt {3}\, c_{1} \right ) t}{6}+\frac {c_{1}^{2}}{4}\right \}, \left \{y \left (t \right ) = -\frac {-2 t^{2} \left (\frac {d}{d t}x \left (t \right )\right )-2 t^{3}-6 x \left (t \right ) \left (\frac {d}{d t}x \left (t \right )\right )-7 x \left (t \right ) t}{9 a}\right \}\right ] \\
\end{align*}
✓ Mathematica. Time used: 0.01 (sec). Leaf size: 28
ode={D[x[t],t]^2+t*D[x[t],t]+a*D[y[t],t]-x[t]==0,D[x[t],t]*D[y[t],t]+t*D[y[t],t]-y[t]==0};
ic={};
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
\begin{align*}
x(t)\to a c_2+c_1 t+c_1{}^2 \\
y(t)\to c_2 (t+c_1) \\
\end{align*}
✓ Sympy. Time used: 1.587 (sec). Leaf size: 20
from sympy import *
t = symbols("t")
a = symbols("a")
x = Function("x")
y = Function("y")
ode=[Eq(a*Derivative(y(t), t) + t*Derivative(x(t), t) - x(t) + Derivative(x(t), t)**2,0),Eq(t*Derivative(y(t), t) - y(t) + Derivative(x(t), t)*Derivative(y(t), t),0)]
ics = {}
dsolve(ode,func=[x(t),y(t)],ics=ics)
\[
\left \{x{\left (t \right )} = C_{1}^{2} + C_{1} t + C_{2} a, y{\left (t \right )} = C_{1} C_{2} + C_{2} t\right \}
\]