Internal problem ID [12559]
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.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_separable]
\[ \boxed {y^{\prime }-\frac {4 t}{1+3 y^{2}}=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 373
dsolve(diff(y(t),t)=4*t/(1+3*y(t)^2),y(t), singsol=all)
\begin{align*} y \left (t \right ) = \frac {\left (27 t^{2}+54 c_{1} +3 \sqrt {81 t^{4}+324 c_{1} t^{2}+324 c_{1}^{2}+3}\right )^{\frac {1}{3}}}{3}-\frac {1}{\left (27 t^{2}+54 c_{1} +3 \sqrt {81 t^{4}+324 c_{1} t^{2}+324 c_{1}^{2}+3}\right )^{\frac {1}{3}}} y \left (t \right ) = -\frac {\left (27 t^{2}+54 c_{1} +3 \sqrt {81 t^{4}+324 c_{1} t^{2}+324 c_{1}^{2}+3}\right )^{\frac {1}{3}}}{6}+\frac {1}{2 \left (27 t^{2}+54 c_{1} +3 \sqrt {81 t^{4}+324 c_{1} t^{2}+324 c_{1}^{2}+3}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (27 t^{2}+54 c_{1} +3 \sqrt {81 t^{4}+324 c_{1} t^{2}+324 c_{1}^{2}+3}\right )^{\frac {1}{3}}}{3}+\frac {1}{\left (27 t^{2}+54 c_{1} +3 \sqrt {81 t^{4}+324 c_{1} t^{2}+324 c_{1}^{2}+3}\right )^{\frac {1}{3}}}\right )}{2} y \left (t \right ) = -\frac {\left (27 t^{2}+54 c_{1} +3 \sqrt {81 t^{4}+324 c_{1} t^{2}+324 c_{1}^{2}+3}\right )^{\frac {1}{3}}}{6}+\frac {1}{2 \left (27 t^{2}+54 c_{1} +3 \sqrt {81 t^{4}+324 c_{1} t^{2}+324 c_{1}^{2}+3}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (27 t^{2}+54 c_{1} +3 \sqrt {81 t^{4}+324 c_{1} t^{2}+324 c_{1}^{2}+3}\right )^{\frac {1}{3}}}{3}+\frac {1}{\left (27 t^{2}+54 c_{1} +3 \sqrt {81 t^{4}+324 c_{1} t^{2}+324 c_{1}^{2}+3}\right )^{\frac {1}{3}}}\right )}{2} \end{align*}
✓ Solution by Mathematica
Time used: 3.132 (sec). Leaf size: 298
DSolve[y'[t]==4*t/(1+3*y[t]^2),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*}