67.2.10 problem Problem 1(j)
Internal
problem
ID
[13967]
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)
Date
solved
:
Tuesday, January 28, 2025 at 06:10:33 AM
CAS
classification
:
[`y=_G(x,y')`]
\begin{align*} \sinh \left (x \right ) {y^{\prime }}^{2}+3 y&=0 \end{align*}
✓ Solution by Maple
Time used: 0.172 (sec). Leaf size: 707
dsolve(sinh(x)*diff(y(x),x)^2+3*y(x)=0,y(x), singsol=all)
\begin{align*}
y &= 0 \\
y &= -\frac {{\operatorname {RootOf}\left (-\operatorname {JacobiSN}\left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-{\mathrm e}^{x}, \operatorname {index} =1\right ) \sqrt {1-{\mathrm e}^{x}}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \,{\mathrm e}^{x}-2, \operatorname {index} =1\right ) \left (\sqrt {2}\, \sqrt {3}\, \sqrt {-{\mathrm e}^{x} \left ({\mathrm e}^{2 x}-1\right )}\, c_{1} -2 \textit {\_Z} \right ) {\mathrm e}^{-x}}{12 \,{\mathrm e}^{2 x}-12}, \frac {\sqrt {2}}{2}\right )+\sqrt {1-{\mathrm e}^{x}}\right )}^{2} {\mathrm e}^{-x}}{6 \,{\mathrm e}^{2 x}-6} \\
y &= -\frac {{\operatorname {RootOf}\left (\operatorname {JacobiSN}\left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-{\mathrm e}^{x}, \operatorname {index} =1\right ) \sqrt {1-{\mathrm e}^{x}}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \,{\mathrm e}^{x}-2, \operatorname {index} =1\right ) \left (\sqrt {2}\, \sqrt {3}\, \sqrt {-{\mathrm e}^{x} \left ({\mathrm e}^{2 x}-1\right )}\, c_{1} +2 \textit {\_Z} \right ) {\mathrm e}^{-x}}{12 \,{\mathrm e}^{2 x}-12}, \frac {\sqrt {2}}{2}\right )+\sqrt {1-{\mathrm e}^{x}}\right )}^{2} {\mathrm e}^{-x}}{6 \,{\mathrm e}^{2 x}-6} \\
y &= -\frac {{\operatorname {RootOf}\left (\operatorname {JacobiSN}\left (\frac {\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 ) \left (3 \,{\mathrm e}^{2 x} \operatorname {RootOf}\left (-1+\left (-6 \,{\mathrm e}^{3 x}+6 \,{\mathrm e}^{x}\right ) \textit {\_Z}^{2}\right ) c_{1} -3 c_{1} \operatorname {RootOf}\left (-1+\left (-6 \,{\mathrm e}^{3 x}+6 \,{\mathrm e}^{x}\right ) \textit {\_Z}^{2}\right )-\textit {\_Z} \,{\mathrm e}^{-x}\right )}{6 \sqrt {1-{\mathrm e}^{x}}\, \left ({\mathrm e}^{x}+1\right )}, \frac {\sqrt {2}}{2}\right )+\sqrt {1-{\mathrm e}^{x}}\right )}^{2} {\mathrm e}^{-x}}{6 \,{\mathrm e}^{2 x}-6} \\
y &= -\frac {{\operatorname {RootOf}\left (\operatorname {JacobiSN}\left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-{\mathrm e}^{x}, \operatorname {index} =1\right ) \sqrt {1-{\mathrm e}^{x}}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \,{\mathrm e}^{x}-2, \operatorname {index} =1\right ) \left (\sqrt {2}\, \sqrt {3}\, \sqrt {-{\mathrm e}^{x} \left ({\mathrm e}^{2 x}-1\right )}\, c_{1} -2 \textit {\_Z} \right ) {\mathrm e}^{-x}}{12 \,{\mathrm e}^{2 x}-12}, \frac {\sqrt {2}}{2}\right )+\sqrt {1-{\mathrm e}^{x}}\right )}^{2} {\mathrm e}^{-x}}{6 \,{\mathrm e}^{2 x}-6} \\
y &= -\frac {{\operatorname {RootOf}\left (-\operatorname {JacobiSN}\left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-{\mathrm e}^{x}, \operatorname {index} =1\right ) \sqrt {1-{\mathrm e}^{x}}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \,{\mathrm e}^{x}-2, \operatorname {index} =1\right ) \left (\sqrt {2}\, \sqrt {3}\, \sqrt {-{\mathrm e}^{x} \left ({\mathrm e}^{2 x}-1\right )}\, c_{1} +2 \textit {\_Z} \right ) {\mathrm e}^{-x}}{12 \,{\mathrm e}^{2 x}-12}, \frac {\sqrt {2}}{2}\right )+\sqrt {1-{\mathrm e}^{x}}\right )}^{2} {\mathrm e}^{-x}}{6 \,{\mathrm e}^{2 x}-6} \\
y &= -\frac {{\operatorname {RootOf}\left (-\operatorname {JacobiSN}\left (\frac {\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 ) \left (3 \,{\mathrm e}^{2 x} \operatorname {RootOf}\left (-1+\left (-6 \,{\mathrm e}^{3 x}+6 \,{\mathrm e}^{x}\right ) \textit {\_Z}^{2}\right ) c_{1} -3 c_{1} \operatorname {RootOf}\left (-1+\left (-6 \,{\mathrm e}^{3 x}+6 \,{\mathrm e}^{x}\right ) \textit {\_Z}^{2}\right )-\textit {\_Z} \,{\mathrm e}^{-x}\right )}{6 \sqrt {1-{\mathrm e}^{x}}\, \left ({\mathrm e}^{x}+1\right )}, \frac {\sqrt {2}}{2}\right )+\sqrt {1-{\mathrm e}^{x}}\right )}^{2} {\mathrm e}^{-x}}{6 \,{\mathrm e}^{2 x}-6} \\
\end{align*}
✓ Solution by Mathematica
Time used: 0.359 (sec). Leaf size: 145
DSolve[Sinh[x]*D[y[x],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*}