72.1.15 problem 18

Internal problem ID [14615]
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 : Tuesday, January 28, 2025 at 06:44:38 AM
CAS classification : [_separable]

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

✓ Solution by Maple

Time used: 0.001 (sec). Leaf size: 278

dsolve(diff(y(t),t)=4*t/(1+3*y(t)^2),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*}

✓ Solution by Mathematica

Time used: 2.315 (sec). Leaf size: 298

DSolve[D[y[t],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*}