Internal problem ID [9936]
Internal file name [OUTPUT/8883_Monday_June_06_2022_05_45_01_AM_85178070/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1614 (6.24).
ODE order: 2.
ODE degree: 1.
The type(s) of ODE detected by this program : "second_order_ode_missing_x"
Maple gives the following as the ode type
[[_2nd_order, _missing_x]]
Unable to solve or complete the solution.
\[ \boxed {y^{\prime \prime }+3 a y^{\prime }-2 y^{3}+2 a^{2} y=0} \]
This is missing independent variable second order ode. Solved by reduction of order by using substitution which makes the dependent variable \(y\) an independent variable. Using \begin {align*} y' &= p(y) \end {align*}
Then \begin {align*} y'' &= \frac {dp}{dx}\\ &= \frac {dy}{dx} \frac {dp}{dy}\\ &= p \frac {dp}{dy} \end {align*}
Hence the ode becomes \begin {align*} p \left (y \right ) \left (\frac {d}{d y}p \left (y \right )\right )+3 a p \left (y \right )+\left (2 a^{2}-2 y^{2}\right ) y = 0 \end {align*}
Which is now solved as first order ode for \(p(y)\). Unable to determine ODE type.
Unable to solve. Terminating
Maple trace
`Methods for second order ODEs: --- Trying classification methods --- trying 2nd order Liouville trying 2nd order WeierstrassP trying 2nd order JacobiSN <- 2nd_order JacobiSN successful`
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 34
dsolve(diff(diff(y(x),x),x)+3*a*diff(y(x),x)-2*y(x)^3+2*a^2*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = -\operatorname {JacobiSN}\left (\frac {\left (-c_{1} a +\sqrt {-{\mathrm e}^{-2 a x}}\right ) c_{2}}{a}, i\right ) c_{2} {\mathrm e}^{-a x} \]
✓ Solution by Mathematica
Time used: 3.421 (sec). Leaf size: 32
DSolve[2*a^2*y[x] - 2*y[x]^3 + 3*a*y'[x] + y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to -i a c_1 e^{-a x} \text {sn}\left (\left .e^{-a x} c_1+c_2\right |-1\right ) \]