74.4.22 problem 22
Internal
problem
ID
[15836]
Book
:
INTRODUCTORY
DIFFERENTIAL
EQUATIONS.
Martha
L.
Abell,
James
P.
Braselton.
Fourth
edition
2014.
ElScAe.
2014
Section
:
Chapter
2.
First
Order
Equations.
Exercises
2.2,
page
39
Problem
number
:
22
Date
solved
:
Thursday, March 13, 2025 at 06:48:43 AM
CAS
classification
:
[[_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]
\begin{align*} y^{\prime }&=\frac {t^{3}}{y \sqrt {\left (1-y^{2}\right ) \left (t^{4}+9\right )}} \end{align*}
✓ Maple. Time used: 0.006 (sec). Leaf size: 50
ode:=diff(y(t),t) = t^3/y(t)/((1-y(t)^2)*(t^4+9))^(1/2);
dsolve(ode,y(t), singsol=all);
\[
-\left (-\frac {y^{2}}{3}+\int _{}^{t}\frac {\textit {\_a}^{3}}{\sqrt {-\left (\textit {\_a}^{4}+9\right ) \left (-1+y^{2}\right )}}d \textit {\_a} +\frac {1}{3}\right ) \sqrt {y+1}\, \sqrt {y-1}+c_{1} = 0
\]
✓ Mathematica. Time used: 2.894 (sec). Leaf size: 519
ode=D[y[t],t]==t^3/(y[t]*Sqrt[(1-y[t]^2)*(t^4+9)]);
ic={};
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
\begin{align*}
y(t)\to -\sqrt {1+\left (\frac {3}{2}\right )^{2/3} \sqrt [3]{-t^4-4 i c_1 \sqrt {t^4+9}-9+4 c_1{}^2}} \\
y(t)\to \sqrt {1+\left (\frac {3}{2}\right )^{2/3} \sqrt [3]{-t^4-4 i c_1 \sqrt {t^4+9}-9+4 c_1{}^2}} \\
y(t)\to -\frac {1}{2} \sqrt {-\sqrt [3]{2} 3^{2/3} \sqrt [3]{-t^4-4 i c_1 \sqrt {t^4+9}-9+4 c_1{}^2}-3 i \sqrt [3]{2} \sqrt [6]{3} \sqrt [3]{-t^4-4 i c_1 \sqrt {t^4+9}-9+4 c_1{}^2}+4} \\
y(t)\to \frac {1}{2} \sqrt {-\sqrt [3]{2} 3^{2/3} \sqrt [3]{-t^4-4 i c_1 \sqrt {t^4+9}-9+4 c_1{}^2}-3 i \sqrt [3]{2} \sqrt [6]{3} \sqrt [3]{-t^4-4 i c_1 \sqrt {t^4+9}-9+4 c_1{}^2}+4} \\
y(t)\to -\frac {1}{2} \sqrt {-\sqrt [3]{2} 3^{2/3} \sqrt [3]{-t^4-4 i c_1 \sqrt {t^4+9}-9+4 c_1{}^2}+3 i \sqrt [3]{2} \sqrt [6]{3} \sqrt [3]{-t^4-4 i c_1 \sqrt {t^4+9}-9+4 c_1{}^2}+4} \\
y(t)\to \frac {1}{2} \sqrt {-\sqrt [3]{2} 3^{2/3} \sqrt [3]{-t^4-4 i c_1 \sqrt {t^4+9}-9+4 c_1{}^2}+3 i \sqrt [3]{2} \sqrt [6]{3} \sqrt [3]{-t^4-4 i c_1 \sqrt {t^4+9}-9+4 c_1{}^2}+4} \\
\end{align*}
✓ Sympy. Time used: 6.765 (sec). Leaf size: 272
from sympy import *
t = symbols("t")
y = Function("y")
ode = Eq(-t**3/(sqrt((1 - y(t)**2)*(t**4 + 9))*y(t)) + Derivative(y(t), t),0)
ics = {}
dsolve(ode,func=y(t),ics=ics)
\[
\begin {cases} \left (y{\left (t \right )} + 1\right ) \sqrt {\left (y{\left (t \right )} + 1\right )^{2} - 2 y{\left (t \right )} - 2} - \sqrt {\left (y{\left (t \right )} + 1\right )^{2} - 2 y{\left (t \right )} - 2} & \text {for}\: \frac {1}{\left |{y{\left (t \right )} + 1}\right |} < 1 \wedge \left |{y{\left (t \right )} + 1}\right | < 1 \\\left (y{\left (t \right )} + 1\right )^{2} \sqrt {\left (y{\left (t \right )} + 1\right )^{2} - 2 y{\left (t \right )} - 2} - \left (y{\left (t \right )} + 1\right ) \sqrt {\left (y{\left (t \right )} + 1\right )^{2} - 2 y{\left (t \right )} - 2} & \text {for}\: \frac {1}{\left |{y{\left (t \right )} + 1}\right |} < 1 \vee \left |{y{\left (t \right )} + 1}\right | < 1 \\\left (y{\left (t \right )} + 1\right )^{2} \sqrt {\left (y{\left (t \right )} + 1\right )^{2} - 2 y{\left (t \right )} - 2} {G_{2, 2}^{1, 1}\left (\begin {matrix} 0 & 1 \\0 & -1 \end {matrix} \middle | {y{\left (t \right )} + 1} \right )} + \left (y{\left (t \right )} + 1\right )^{2} \sqrt {\left (y{\left (t \right )} + 1\right )^{2} - 2 y{\left (t \right )} - 2} {G_{2, 2}^{0, 2}\left (\begin {matrix} 0, 1 & \\ & -1, 0 \end {matrix} \middle | {y{\left (t \right )} + 1} \right )} - \left (y{\left (t \right )} + 1\right ) \sqrt {\left (y{\left (t \right )} + 1\right )^{2} - 2 y{\left (t \right )} - 2} {G_{2, 2}^{1, 1}\left (\begin {matrix} 0 & 1 \\0 & -1 \end {matrix} \middle | {y{\left (t \right )} + 1} \right )} - \left (y{\left (t \right )} + 1\right ) \sqrt {\left (y{\left (t \right )} + 1\right )^{2} - 2 y{\left (t \right )} - 2} {G_{2, 2}^{0, 2}\left (\begin {matrix} 0, 1 & \\ & -1, 0 \end {matrix} \middle | {y{\left (t \right )} + 1} \right )} & \text {otherwise} \end {cases} + \frac {\sqrt {- t^{4} y^{2}{\left (t \right )} + t^{4} - 9 y^{2}{\left (t \right )} + 9}}{2 \sqrt {y^{2}{\left (t \right )} - 1}} = C_{1}
\]