Internal problem ID [14295]
Book: INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton.
Fourth edition 2014. ElScAe. 2014
Section: Chapter 2. First Order Equations. Exercises 2.4, page 57
Problem number: 7.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_1st_order, `_with_symmetry_[F(x)*G(y),0]`]]
\[ \boxed {\sin \left (2 t \right ) y+\left (\sqrt {y}+\cos \left (2 t \right )\right ) y^{\prime }=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 21
dsolve(y(t)*sin(2*t)+(sqrt(y(t))+cos(2*t))*diff(y(t),t)=0,y(t), singsol=all)
\[ -\frac {\cos \left (2 t \right )}{2 y \left (t \right )^{2}}-\frac {2}{3 y \left (t \right )^{\frac {3}{2}}}+c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 60.322 (sec). Leaf size: 1897
DSolve[y[t]*Sin[2*t]+(Sqrt[y[t]]+Cos[2*t])*y'[t]==0,y[t],t,IncludeSingularSolutions -> True]
\begin{align*} y(t)\to \frac {1}{6} \left (-\sqrt {\frac {6^{2/3} \sqrt [3]{-6 c_1{}^3 \cos ^3(2 t)+\sqrt {2} \sqrt {-c_1{}^4 (9 c_1 \cos (2 t)+3 c_1 \cos (6 t)-2)}+2 c_1{}^2}}{c_1{}^2}+\frac {6 \sqrt [3]{6} \cos ^2(2 t)}{\sqrt [3]{-6 c_1{}^3 \cos ^3(2 t)+\sqrt {2} \sqrt {-c_1{}^4 (9 c_1 \cos (2 t)+3 c_1 \cos (6 t)-2)}+2 c_1{}^2}}-\frac {6 \cos (2 t)}{c_1}}-\sqrt {-\frac {6^{2/3} \sqrt [3]{-6 c_1{}^3 \cos ^3(2 t)+\sqrt {2} \sqrt {-c_1{}^4 (9 c_1 \cos (2 t)+3 c_1 \cos (6 t)-2)}+2 c_1{}^2}}{c_1{}^2}-\frac {6 \sqrt [3]{6} \cos ^2(2 t)}{\sqrt [3]{-6 c_1{}^3 \cos ^3(2 t)+\sqrt {2} \sqrt {-c_1{}^4 (9 c_1 \cos (2 t)+3 c_1 \cos (6 t)-2)}+2 c_1{}^2}}-\frac {24}{c_1{}^2 \sqrt {\frac {6^{2/3} \sqrt [3]{-6 c_1{}^3 \cos ^3(2 t)+\sqrt {2} \sqrt {-c_1{}^4 (9 c_1 \cos (2 t)+3 c_1 \cos (6 t)-2)}+2 c_1{}^2}}{c_1{}^2}+\frac {6 \sqrt [3]{6} \cos ^2(2 t)}{\sqrt [3]{-6 c_1{}^3 \cos ^3(2 t)+\sqrt {2} \sqrt {-c_1{}^4 (9 c_1 \cos (2 t)+3 c_1 \cos (6 t)-2)}+2 c_1{}^2}}-\frac {6 \cos (2 t)}{c_1}}}-\frac {12 \cos (2 t)}{c_1}}\right ) \\ y(t)\to \frac {1}{6} \left (\sqrt {-\frac {6^{2/3} \sqrt [3]{-6 c_1{}^3 \cos ^3(2 t)+\sqrt {2} \sqrt {-c_1{}^4 (9 c_1 \cos (2 t)+3 c_1 \cos (6 t)-2)}+2 c_1{}^2}}{c_1{}^2}-\frac {6 \sqrt [3]{6} \cos ^2(2 t)}{\sqrt [3]{-6 c_1{}^3 \cos ^3(2 t)+\sqrt {2} \sqrt {-c_1{}^4 (9 c_1 \cos (2 t)+3 c_1 \cos (6 t)-2)}+2 c_1{}^2}}-\frac {24}{c_1{}^2 \sqrt {\frac {6^{2/3} \sqrt [3]{-6 c_1{}^3 \cos ^3(2 t)+\sqrt {2} \sqrt {-c_1{}^4 (9 c_1 \cos (2 t)+3 c_1 \cos (6 t)-2)}+2 c_1{}^2}}{c_1{}^2}+\frac {6 \sqrt [3]{6} \cos ^2(2 t)}{\sqrt [3]{-6 c_1{}^3 \cos ^3(2 t)+\sqrt {2} \sqrt {-c_1{}^4 (9 c_1 \cos (2 t)+3 c_1 \cos (6 t)-2)}+2 c_1{}^2}}-\frac {6 \cos (2 t)}{c_1}}}-\frac {12 \cos (2 t)}{c_1}}-\sqrt {\frac {6^{2/3} \sqrt [3]{-6 c_1{}^3 \cos ^3(2 t)+\sqrt {2} \sqrt {-c_1{}^4 (9 c_1 \cos (2 t)+3 c_1 \cos (6 t)-2)}+2 c_1{}^2}}{c_1{}^2}+\frac {6 \sqrt [3]{6} \cos ^2(2 t)}{\sqrt [3]{-6 c_1{}^3 \cos ^3(2 t)+\sqrt {2} \sqrt {-c_1{}^4 (9 c_1 \cos (2 t)+3 c_1 \cos (6 t)-2)}+2 c_1{}^2}}-\frac {6 \cos (2 t)}{c_1}}\right ) \\ y(t)\to \frac {1}{6} \left (\sqrt {\frac {6^{2/3} \sqrt [3]{-6 c_1{}^3 \cos ^3(2 t)+\sqrt {2} \sqrt {-c_1{}^4 (9 c_1 \cos (2 t)+3 c_1 \cos (6 t)-2)}+2 c_1{}^2}}{c_1{}^2}+\frac {6 \sqrt [3]{6} \cos ^2(2 t)}{\sqrt [3]{-6 c_1{}^3 \cos ^3(2 t)+\sqrt {2} \sqrt {-c_1{}^4 (9 c_1 \cos (2 t)+3 c_1 \cos (6 t)-2)}+2 c_1{}^2}}-\frac {6 \cos (2 t)}{c_1}}-\sqrt {-\frac {6^{2/3} \sqrt [3]{-6 c_1{}^3 \cos ^3(2 t)+\sqrt {2} \sqrt {-c_1{}^4 (9 c_1 \cos (2 t)+3 c_1 \cos (6 t)-2)}+2 c_1{}^2}}{c_1{}^2}-\frac {6 \sqrt [3]{6} \cos ^2(2 t)}{\sqrt [3]{-6 c_1{}^3 \cos ^3(2 t)+\sqrt {2} \sqrt {-c_1{}^4 (9 c_1 \cos (2 t)+3 c_1 \cos (6 t)-2)}+2 c_1{}^2}}+\frac {24}{c_1{}^2 \sqrt {\frac {6^{2/3} \sqrt [3]{-6 c_1{}^3 \cos ^3(2 t)+\sqrt {2} \sqrt {-c_1{}^4 (9 c_1 \cos (2 t)+3 c_1 \cos (6 t)-2)}+2 c_1{}^2}}{c_1{}^2}+\frac {6 \sqrt [3]{6} \cos ^2(2 t)}{\sqrt [3]{-6 c_1{}^3 \cos ^3(2 t)+\sqrt {2} \sqrt {-c_1{}^4 (9 c_1 \cos (2 t)+3 c_1 \cos (6 t)-2)}+2 c_1{}^2}}-\frac {6 \cos (2 t)}{c_1}}}-\frac {12 \cos (2 t)}{c_1}}\right ) \\ y(t)\to \frac {1}{6} \left (\sqrt {\frac {6^{2/3} \sqrt [3]{-6 c_1{}^3 \cos ^3(2 t)+\sqrt {2} \sqrt {-c_1{}^4 (9 c_1 \cos (2 t)+3 c_1 \cos (6 t)-2)}+2 c_1{}^2}}{c_1{}^2}+\frac {6 \sqrt [3]{6} \cos ^2(2 t)}{\sqrt [3]{-6 c_1{}^3 \cos ^3(2 t)+\sqrt {2} \sqrt {-c_1{}^4 (9 c_1 \cos (2 t)+3 c_1 \cos (6 t)-2)}+2 c_1{}^2}}-\frac {6 \cos (2 t)}{c_1}}+\sqrt {-\frac {6^{2/3} \sqrt [3]{-6 c_1{}^3 \cos ^3(2 t)+\sqrt {2} \sqrt {-c_1{}^4 (9 c_1 \cos (2 t)+3 c_1 \cos (6 t)-2)}+2 c_1{}^2}}{c_1{}^2}-\frac {6 \sqrt [3]{6} \cos ^2(2 t)}{\sqrt [3]{-6 c_1{}^3 \cos ^3(2 t)+\sqrt {2} \sqrt {-c_1{}^4 (9 c_1 \cos (2 t)+3 c_1 \cos (6 t)-2)}+2 c_1{}^2}}+\frac {24}{c_1{}^2 \sqrt {\frac {6^{2/3} \sqrt [3]{-6 c_1{}^3 \cos ^3(2 t)+\sqrt {2} \sqrt {-c_1{}^4 (9 c_1 \cos (2 t)+3 c_1 \cos (6 t)-2)}+2 c_1{}^2}}{c_1{}^2}+\frac {6 \sqrt [3]{6} \cos ^2(2 t)}{\sqrt [3]{-6 c_1{}^3 \cos ^3(2 t)+\sqrt {2} \sqrt {-c_1{}^4 (9 c_1 \cos (2 t)+3 c_1 \cos (6 t)-2)}+2 c_1{}^2}}-\frac {6 \cos (2 t)}{c_1}}}-\frac {12 \cos (2 t)}{c_1}}\right ) \\ \end{align*}