Internal problem ID [3380]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 5
Problem number: 122.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_separable]
\[ \boxed {y^{\prime }-\sec \left (x \right )^{2} \sec \left (y\right )^{3}=0} \]
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 74
dsolve(diff(y(x),x) = sec(x)^2*sec(y(x))^3,y(x), singsol=all)
\[ y \left (x \right ) = \arctan \left (\frac {3 c_{1} +3 \tan \left (x \right )}{\operatorname {RootOf}\left (\textit {\_Z}^{6}+3 \textit {\_Z}^{4}+9 c_{1}^{2}+18 c_{1} \tan \left (x \right )+9 \tan \left (x \right )^{2}-4\right )^{2}+2}, \operatorname {RootOf}\left (\textit {\_Z}^{6}+3 \textit {\_Z}^{4}+9 c_{1}^{2}+18 c_{1} \tan \left (x \right )+9 \tan \left (x \right )^{2}-4\right )\right ) \]
✓ Solution by Mathematica
Time used: 24.108 (sec). Leaf size: 478
DSolve[y'[x]==Sec[x]^2 Sec[y[x]]^3,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \arcsin \left (\frac {\sqrt [3]{-3 \tan (x)+\sqrt {9 \tan ^2(x)+18 c_1 \tan (x)-4+9 c_1{}^2}-3 c_1}}{\sqrt [3]{2}}+\frac {\sqrt [3]{2}}{\sqrt [3]{-3 \tan (x)+\sqrt {9 \tan ^2(x)+18 c_1 \tan (x)-4+9 c_1{}^2}-3 c_1}}\right ) y(x)\to -\arcsin \left (\frac {\left (1-i \sqrt {3}\right ) \sqrt [3]{-3 \tan (x)+\sqrt {9 \tan ^2(x)+18 c_1 \tan (x)-4+9 c_1{}^2}-3 c_1}}{2 \sqrt [3]{2}}+\frac {1+i \sqrt {3}}{2^{2/3} \sqrt [3]{-3 \tan (x)+\sqrt {9 \tan ^2(x)+18 c_1 \tan (x)-4+9 c_1{}^2}-3 c_1}}\right ) y(x)\to -\arcsin \left (\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{-3 \tan (x)+\sqrt {9 \tan ^2(x)+18 c_1 \tan (x)-4+9 c_1{}^2}-3 c_1}}{2 \sqrt [3]{2}}+\frac {1-i \sqrt {3}}{2^{2/3} \sqrt [3]{-3 \tan (x)+\sqrt {9 \tan ^2(x)+18 c_1 \tan (x)-4+9 c_1{}^2}-3 c_1}}\right ) y(x)\to \arcsin \left (\frac {\sqrt [3]{\sqrt {9 \tan ^2(x)-4}-3 \tan (x)}}{\sqrt [3]{2}}+\frac {\sqrt [3]{2}}{\sqrt [3]{\sqrt {9 \tan ^2(x)-4}-3 \tan (x)}}\right ) y(x)\to -\arcsin \left (\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{\sqrt {9 \tan ^2(x)-4}-3 \tan (x)}}{2 \sqrt [3]{2}}+\frac {1-i \sqrt {3}}{2^{2/3} \sqrt [3]{\sqrt {9 \tan ^2(x)-4}-3 \tan (x)}}\right ) \end{align*}