66.1.15 problem 18

Internal problem ID [15902]
Book : DIFFERENTIAL EQUATIONS by Paul Blanchard, Robert L. Devaney, Glen R. Hall. 4th edition. Brooks/Cole. Boston, USA. 2012
Section : Chapter 1. First-Order Differential Equations. Exercises section 1.2. page 33
Problem number : 18
Date solved : Thursday, October 02, 2025 at 10:29:34 AM
CAS classification : [_separable]

\begin{align*} y^{\prime }&=\frac {4 t}{1+3 y^{2}} \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 278

ode:=diff(y(t),t) = 4*t/(1+3*y(t)^2); 
dsolve(ode,y(t), singsol=all);
 

\begin{align*} y &= \frac {\left (27 t^{2}+54 c_1 +3 \sqrt {81 t^{4}+324 c_1 \,t^{2}+324 c_1^{2}+3}\right )^{{2}/{3}}-3}{3 \left (27 t^{2}+54 c_1 +3 \sqrt {81 t^{4}+324 c_1 \,t^{2}+324 c_1^{2}+3}\right )^{{1}/{3}}} \\ y &= -\frac {\left (1+i \sqrt {3}\right ) \left (27 t^{2}+54 c_1 +3 \sqrt {81 t^{4}+324 c_1 \,t^{2}+324 c_1^{2}+3}\right )^{{2}/{3}}+3 i \sqrt {3}-3}{6 \left (27 t^{2}+54 c_1 +3 \sqrt {81 t^{4}+324 c_1 \,t^{2}+324 c_1^{2}+3}\right )^{{1}/{3}}} \\ y &= \frac {i \sqrt {3}\, \left (27 t^{2}+54 c_1 +3 \sqrt {81 t^{4}+324 c_1 \,t^{2}+324 c_1^{2}+3}\right )^{{2}/{3}}+3 i \sqrt {3}-\left (27 t^{2}+54 c_1 +3 \sqrt {81 t^{4}+324 c_1 \,t^{2}+324 c_1^{2}+3}\right )^{{2}/{3}}+3}{6 \left (27 t^{2}+54 c_1 +3 \sqrt {81 t^{4}+324 c_1 \,t^{2}+324 c_1^{2}+3}\right )^{{1}/{3}}} \\ \end{align*}

✓ Mathematica. Time used: 1.337 (sec). Leaf size: 298

ode=D[y[t],t]==4*t/(1+3*y[t]^2); 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to \frac {\sqrt [3]{54 t^2+\sqrt {108+729 \left (2 t^2+c_1\right ){}^2}+27 c_1}}{3 \sqrt [3]{2}}-\frac {\sqrt [3]{2}}{\sqrt [3]{54 t^2+\sqrt {108+729 \left (2 t^2+c_1\right ){}^2}+27 c_1}}\\ y(t)&\to \frac {\left (-1+i \sqrt {3}\right ) \sqrt [3]{54 t^2+\sqrt {108+729 \left (2 t^2+c_1\right ){}^2}+27 c_1}}{6 \sqrt [3]{2}}+\frac {1+i \sqrt {3}}{2^{2/3} \sqrt [3]{54 t^2+\sqrt {108+729 \left (2 t^2+c_1\right ){}^2}+27 c_1}}\\ y(t)&\to \frac {1-i \sqrt {3}}{2^{2/3} \sqrt [3]{54 t^2+\sqrt {108+729 \left (2 t^2+c_1\right ){}^2}+27 c_1}}-\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{54 t^2+\sqrt {108+729 \left (2 t^2+c_1\right ){}^2}+27 c_1}}{6 \sqrt [3]{2}} \end{align*}

✗ Sympy

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-4*t/(3*y(t)**2 + 1) + Derivative(y(t), t),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics)
 

Timed Out