74.7.56 problem 61
Internal
problem
ID
[16062]
Book
:
INTRODUCTORY
DIFFERENTIAL
EQUATIONS.
Martha
L.
Abell,
James
P.
Braselton.
Fourth
edition
2014.
ElScAe.
2014
Section
:
Chapter
2.
First
Order
Equations.
Exercises
2.5,
page
64
Problem
number
:
61
Date
solved
:
Monday, March 31, 2025 at 02:37:31 PM
CAS
classification
:
[[_1st_order, _with_linear_symmetries], _dAlembert]
\begin{align*} y&=t \left (2-y^{\prime }\right )+2 {y^{\prime }}^{2}+1 \end{align*}
✓ Maple. Time used: 0.029 (sec). Leaf size: 613
ode:=y(t) = t*(2-diff(y(t),t))+2*diff(y(t),t)^2+1;
dsolve(ode,y(t), singsol=all);
\begin{align*}
y &= -\frac {\left (\frac {t -4}{\left (-3 c_1 +\sqrt {-t^{3}+9 c_1^{2}+12 t^{2}-48 t +64}\right )^{{1}/{3}}}+\left (-3 c_1 +\sqrt {-t^{3}+9 c_1^{2}+12 t^{2}-48 t +64}\right )^{{1}/{3}}-2\right ) \left (\frac {t -4}{\left (-3 c_1 +\sqrt {-t^{3}+9 c_1^{2}+12 t^{2}-48 t +64}\right )^{{1}/{3}}}+\left (-3 c_1 +\sqrt {-t^{3}+9 c_1^{2}+12 t^{2}-48 t +64}\right )^{{1}/{3}}+2\right ) t}{4}+\frac {\left (\frac {\left (t -4\right )^{2}}{\left (-3 c_1 +\sqrt {-t^{3}+9 c_1^{2}+12 t^{2}-48 t +64}\right )^{{2}/{3}}}+2 t -4+\left (-3 c_1 +\sqrt {-t^{3}+9 c_1^{2}+12 t^{2}-48 t +64}\right )^{{2}/{3}}\right )^{2}}{8}+1 \\
y &= -\frac {2 \left (1+i \sqrt {3}\right ) \left (t -4\right )^{3} \left (-3 c_1 +\sqrt {-t^{3}+9 c_1^{2}+12 t^{2}-48 t +64}\right )^{{2}/{3}}+\left (-4 t^{2}+16 t -112\right ) \left (-3 c_1 +\sqrt {-t^{3}+9 c_1^{2}+12 t^{2}-48 t +64}\right )^{{4}/{3}}+\left (1+i \sqrt {3}\right ) \left (-3 c_1 +\sqrt {-t^{3}+9 c_1^{2}+12 t^{2}-48 t +64}\right )^{{8}/{3}}+\left (i \sqrt {3}-1\right ) \left (t^{3}-12 t^{2}-36 c_1^{2}+12 c_1 \sqrt {-t^{3}+9 c_1^{2}+12 t^{2}-48 t +64}+48 t -64\right ) \left (t -4\right )}{16 \left (-3 c_1 +\sqrt {-t^{3}+9 c_1^{2}+12 t^{2}-48 t +64}\right )^{{4}/{3}}} \\
y &= \frac {2 \left (i \sqrt {3}-1\right ) \left (t -4\right )^{3} \left (-3 c_1 +\sqrt {-t^{3}+9 c_1^{2}+12 t^{2}-48 t +64}\right )^{{2}/{3}}+4 \left (t^{2}-4 t +28\right ) \left (-3 c_1 +\sqrt {-t^{3}+9 c_1^{2}+12 t^{2}-48 t +64}\right )^{{4}/{3}}+\left (i \sqrt {3}-1\right ) \left (-3 c_1 +\sqrt {-t^{3}+9 c_1^{2}+12 t^{2}-48 t +64}\right )^{{8}/{3}}+\left (1+i \sqrt {3}\right ) \left (t^{3}-12 t^{2}-36 c_1^{2}+12 c_1 \sqrt {-t^{3}+9 c_1^{2}+12 t^{2}-48 t +64}+48 t -64\right ) \left (t -4\right )}{16 \left (-3 c_1 +\sqrt {-t^{3}+9 c_1^{2}+12 t^{2}-48 t +64}\right )^{{4}/{3}}} \\
\end{align*}
✓ Mathematica. Time used: 61.106 (sec). Leaf size: 2369
ode=y[t]==t*(2-D[y[t],t])+(2*D[y[t],t]^2+1);
ic={};
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
\begin{align*} \text {Solution too large to show}\end{align*}
✗ Sympy
from sympy import *
t = symbols("t")
y = Function("y")
ode = Eq(-t*(2 - Derivative(y(t), t)) + y(t) - 2*Derivative(y(t), t)**2 - 1,0)
ics = {}
dsolve(ode,func=y(t),ics=ics)
Timed Out