Internal problem ID [10883]
Book: APPLIED DIFFERENTIAL EQUATIONS The Primary Course by Vladimir A. Dobrushkin.
CRC Press 2015
Section: Chapter 4, Second and Higher Order Linear Differential Equations. Problems page
221
Problem number: Problem 1(j).
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [`y=_G(x,y')`]
\[ \boxed {\sinh \left (x \right ) {y^{\prime }}^{2}+3 y=0} \]
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 799
dsolve(sinh(x)*diff(y(x),x)^2+3*y(x)=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) = 0 \\ y \left (x \right ) = -\frac {{\mathrm e}^{-x} {\operatorname {RootOf}\left (-\operatorname {JacobiSN}\left (\frac {\left (-\frac {3 \,{\mathrm e}^{3 x} c_{1}}{\sqrt {-6 \,{\mathrm e}^{3 x}+6 \,{\mathrm e}^{x}}}+\frac {3 \,{\mathrm e}^{x} c_{1}}{\sqrt {-6 \,{\mathrm e}^{3 x}+6 \,{\mathrm e}^{x}}}-\textit {\_Z} \right ) \sqrt {-{\mathrm e}^{x}+1}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \,{\mathrm e}^{x}-2, \operatorname {index} =1\right ) \operatorname {RootOf}\left (\textit {\_Z}^{2}-{\mathrm e}^{x}, \operatorname {index} =1\right ) {\mathrm e}^{-x}}{6 \left ({\mathrm e}^{x}-1\right ) \left ({\mathrm e}^{x}+1\right )}, \frac {\sqrt {2}}{2}\right )+\sqrt {-{\mathrm e}^{x}+1}\right )}^{2}}{6 \left ({\mathrm e}^{2 x}-1\right )} \\ y \left (x \right ) = -\frac {{\mathrm e}^{-x} {\operatorname {RootOf}\left (-\operatorname {JacobiSN}\left (\frac {\left (\frac {3 \,{\mathrm e}^{3 x} c_{1}}{\sqrt {-6 \,{\mathrm e}^{3 x}+6 \,{\mathrm e}^{x}}}-\frac {3 \,{\mathrm e}^{x} c_{1}}{\sqrt {-6 \,{\mathrm e}^{3 x}+6 \,{\mathrm e}^{x}}}-\textit {\_Z} \right ) \sqrt {-{\mathrm e}^{x}+1}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \,{\mathrm e}^{x}-2, \operatorname {index} =1\right ) \operatorname {RootOf}\left (\textit {\_Z}^{2}-{\mathrm e}^{x}, \operatorname {index} =1\right ) {\mathrm e}^{-x}}{6 \left ({\mathrm e}^{x}-1\right ) \left ({\mathrm e}^{x}+1\right )}, \frac {\sqrt {2}}{2}\right )+\sqrt {-{\mathrm e}^{x}+1}\right )}^{2}}{6 \left ({\mathrm e}^{2 x}-1\right )} \\ y \left (x \right ) = -\frac {{\mathrm e}^{-x} {\operatorname {RootOf}\left (-\operatorname {JacobiSN}\left (\frac {\left (3 \,{\mathrm e}^{3 x} \operatorname {RootOf}\left (\left (6 \,{\mathrm e}^{3 x}-6 \,{\mathrm e}^{x}\right ) \textit {\_Z}^{2}+1\right ) c_{1} -3 \,{\mathrm e}^{x} \operatorname {RootOf}\left (\left (6 \,{\mathrm e}^{3 x}-6 \,{\mathrm e}^{x}\right ) \textit {\_Z}^{2}+1\right ) c_{1} -\textit {\_Z} \right ) \sqrt {-{\mathrm e}^{x}+1}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \,{\mathrm e}^{x}-2, \operatorname {index} =1\right ) \operatorname {RootOf}\left (\textit {\_Z}^{2}-{\mathrm e}^{x}, \operatorname {index} =1\right ) {\mathrm e}^{-x}}{6 \left ({\mathrm e}^{x}-1\right ) \left ({\mathrm e}^{x}+1\right )}, \frac {\sqrt {2}}{2}\right )+\sqrt {-{\mathrm e}^{x}+1}\right )}^{2}}{6 \left ({\mathrm e}^{2 x}-1\right )} \\ y \left (x \right ) = -\frac {{\mathrm e}^{-x} {\operatorname {RootOf}\left (\operatorname {JacobiSN}\left (\frac {\left (-\frac {3 \,{\mathrm e}^{3 x} c_{1}}{\sqrt {-6 \,{\mathrm e}^{3 x}+6 \,{\mathrm e}^{x}}}+\frac {3 \,{\mathrm e}^{x} c_{1}}{\sqrt {-6 \,{\mathrm e}^{3 x}+6 \,{\mathrm e}^{x}}}-\textit {\_Z} \right ) \sqrt {-{\mathrm e}^{x}+1}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \,{\mathrm e}^{x}-2, \operatorname {index} =1\right ) \operatorname {RootOf}\left (\textit {\_Z}^{2}-{\mathrm e}^{x}, \operatorname {index} =1\right ) {\mathrm e}^{-x}}{6 \left ({\mathrm e}^{x}-1\right ) \left ({\mathrm e}^{x}+1\right )}, \frac {\sqrt {2}}{2}\right )+\sqrt {-{\mathrm e}^{x}+1}\right )}^{2}}{6 \left ({\mathrm e}^{2 x}-1\right )} \\ y \left (x \right ) = -\frac {{\mathrm e}^{-x} {\operatorname {RootOf}\left (\operatorname {JacobiSN}\left (\frac {\left (\frac {3 \,{\mathrm e}^{3 x} c_{1}}{\sqrt {-6 \,{\mathrm e}^{3 x}+6 \,{\mathrm e}^{x}}}-\frac {3 \,{\mathrm e}^{x} c_{1}}{\sqrt {-6 \,{\mathrm e}^{3 x}+6 \,{\mathrm e}^{x}}}-\textit {\_Z} \right ) \sqrt {-{\mathrm e}^{x}+1}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \,{\mathrm e}^{x}-2, \operatorname {index} =1\right ) \operatorname {RootOf}\left (\textit {\_Z}^{2}-{\mathrm e}^{x}, \operatorname {index} =1\right ) {\mathrm e}^{-x}}{6 \left ({\mathrm e}^{x}-1\right ) \left ({\mathrm e}^{x}+1\right )}, \frac {\sqrt {2}}{2}\right )+\sqrt {-{\mathrm e}^{x}+1}\right )}^{2}}{6 \left ({\mathrm e}^{2 x}-1\right )} \\ y \left (x \right ) = -\frac {{\mathrm e}^{-x} {\operatorname {RootOf}\left (\operatorname {JacobiSN}\left (\frac {\left (3 \,{\mathrm e}^{3 x} \operatorname {RootOf}\left (\left (6 \,{\mathrm e}^{3 x}-6 \,{\mathrm e}^{x}\right ) \textit {\_Z}^{2}+1\right ) c_{1} -3 \,{\mathrm e}^{x} \operatorname {RootOf}\left (\left (6 \,{\mathrm e}^{3 x}-6 \,{\mathrm e}^{x}\right ) \textit {\_Z}^{2}+1\right ) c_{1} -\textit {\_Z} \right ) \sqrt {-{\mathrm e}^{x}+1}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \,{\mathrm e}^{x}-2, \operatorname {index} =1\right ) \operatorname {RootOf}\left (\textit {\_Z}^{2}-{\mathrm e}^{x}, \operatorname {index} =1\right ) {\mathrm e}^{-x}}{6 \left ({\mathrm e}^{x}-1\right ) \left ({\mathrm e}^{x}+1\right )}, \frac {\sqrt {2}}{2}\right )+\sqrt {-{\mathrm e}^{x}+1}\right )}^{2}}{6 \left ({\mathrm e}^{2 x}-1\right )} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.367 (sec). Leaf size: 145
DSolve[Sinh[x]*y'[x]^2+3*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to 3 i \operatorname {EllipticF}\left (\frac {1}{4} (\pi -2 i x),2\right )^2-\sqrt {3} c_1 \sqrt {i \sinh (x)} \sqrt {\text {csch}(x)} \operatorname {EllipticF}\left (\frac {1}{4} (\pi -2 i x),2\right )+\frac {c_1{}^2}{4} \\ y(x)\to 3 i \operatorname {EllipticF}\left (\frac {1}{4} (\pi -2 i x),2\right )^2+\sqrt {3} c_1 \sqrt {i \sinh (x)} \sqrt {\text {csch}(x)} \operatorname {EllipticF}\left (\frac {1}{4} (\pi -2 i x),2\right )+\frac {c_1{}^2}{4} \\ y(x)\to 0 \\ \end{align*}