Internal problem ID [9935]
Internal file name [OUTPUT/8882_Monday_June_06_2022_05_44_56_AM_39557389/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1613 (6.23).
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.
\[ \boxed {y^{\prime \prime }+5 a y^{\prime }-6 y^{2}+6 a^{2} y=0} \]
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 )+5 a p \left (y \right )+\left (6 a^{2}-6 y \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 <- 2nd_order WeierstrassP successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 27
dsolve(diff(diff(y(x),x),x)+5*a*diff(y(x),x)-6*y(x)^2+6*a^2*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \operatorname {WeierstrassP}\left (\frac {-c_{1} a +{\mathrm e}^{-a x}}{a}, 0, c_{2}\right ) {\mathrm e}^{-2 a x} \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[6*a^2*y[x] - 6*y[x]^2 + 5*a*y'[x] + y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
Not solved