68.4.10 problem 10

Internal problem ID [17190]
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 : 10
Date solved : Thursday, October 02, 2025 at 01:49:58 PM
CAS classification : [_separable]

\begin{align*} \frac {3}{t^{2}}&=\left (\frac {1}{\sqrt {y}}+\sqrt {y}\right ) y^{\prime } \end{align*}

✓ Maple. Time used: 0.001 (sec). Leaf size: 20

ode:=3/t^2 = (1/y(t)^(1/2)+y(t)^(1/2))*diff(y(t),t); 
dsolve(ode,y(t), singsol=all);
 

\[ -\frac {1}{t}-\frac {2 \sqrt {y}\, \left (y+3\right )}{9}+c_1 = 0 \]

✓ Mathematica. Time used: 5.064 (sec). Leaf size: 445

ode=3/t^2==(1/y[t]^(1/2)+y[t]^(1/2))*D[y[t],t]; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to \frac {\left (-2+\sqrt [3]{\frac {81}{t^2}+3 \sqrt {\frac {(-3+c_1 t){}^2 \left (\left (16+9 c_1{}^2\right ) t^2-54 c_1 t+81\right )}{t^4}}-\frac {54 c_1}{t}+8+9 c_1{}^2}\right ){}^2}{2 \sqrt [3]{\frac {81}{t^2}+3 \sqrt {\frac {(-3+c_1 t){}^2 \left (\left (16+9 c_1{}^2\right ) t^2-54 c_1 t+81\right )}{t^4}}-\frac {54 c_1}{t}+8+9 c_1{}^2}}\\ y(t)&\to \frac {1}{4} i \left (\sqrt {3}+i\right ) \sqrt [3]{\frac {81}{t^2}+3 \sqrt {\frac {(-3+c_1 t){}^2 \left (\left (16+9 c_1{}^2\right ) t^2-54 c_1 t+81\right )}{t^4}}-\frac {54 c_1}{t}+8+9 c_1{}^2}+\frac {-1-i \sqrt {3}}{\sqrt [3]{\frac {81}{t^2}+3 \sqrt {\frac {(-3+c_1 t){}^2 \left (\left (16+9 c_1{}^2\right ) t^2-54 c_1 t+81\right )}{t^4}}-\frac {54 c_1}{t}+8+9 c_1{}^2}}-2\\ y(t)&\to -\frac {1}{4} i \left (\sqrt {3}-i\right ) \sqrt [3]{\frac {81}{t^2}+3 \sqrt {\frac {(-3+c_1 t){}^2 \left (\left (16+9 c_1{}^2\right ) t^2-54 c_1 t+81\right )}{t^4}}-\frac {54 c_1}{t}+8+9 c_1{}^2}+\frac {-1+i \sqrt {3}}{\sqrt [3]{\frac {81}{t^2}+3 \sqrt {\frac {(-3+c_1 t){}^2 \left (\left (16+9 c_1{}^2\right ) t^2-54 c_1 t+81\right )}{t^4}}-\frac {54 c_1}{t}+8+9 c_1{}^2}}-2 \end{align*}

✗ Sympy

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq((-sqrt(y(t)) - 1/sqrt(y(t)))*Derivative(y(t), t) + 3/t**2,0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics)
 

Timed Out