7.31 problem 1621 (6.31)

Internal problem ID [9943]
Internal file name [OUTPUT/8890_Monday_June_06_2022_05_46_04_AM_93471/index.tex]

Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1621 (6.31).
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.

\[ \boxed {y^{\prime \prime }+y y^{\prime }-y^{3}+a y=0} \]

7.31.1 Solving as second order ode missing x ode

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 )+y p \left (y \right )+\left (-y^{2}+a \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 
differential order: 2; trying a linearization to 3rd order 
trying 2nd order ODE linearizable_by_differentiation 
trying 2nd order, 2 integrating factors of the form mu(x,y) 
trying differential order: 2; missing variables 
`, `-> Computing symmetries using: way = 3 
`, `-> Computing symmetries using: way = exp_sym 
-> Calling odsolve with the ODE`, (diff(_b(_a), _a))*_b(_a)+_b(_a)*_a-_a^3+a*_a = 0, _b(_a)`   *** Sublevel 2 *** 
   Methods for first order ODEs: 
   --- Trying classification methods --- 
   trying a quadrature 
   trying 1st order linear 
   trying Bernoulli 
   trying separable 
   trying inverse linear 
   trying homogeneous types: 
   trying Chini 
   differential order: 1; looking for linear symmetries 
   trying exact 
   trying Abel 
   <- Abel successful 
<- differential order: 2; canonical coordinates successful 
<- differential order 2; missing variables successful`
 

✓ Solution by Maple

Time used: 0.016 (sec). Leaf size: 108

dsolve(diff(diff(y(x),x),x)+y(x)*diff(y(x),x)-y(x)^3+a*y(x)=0,y(x), singsol=all)
 

\[ -\frac {\left (\int _{}^{y \left (x \right )}\frac {4 {\operatorname {RootOf}\left (\left (-4 \textit {\_a}^{6}+12 \textit {\_a}^{4} a -12 \textit {\_a}^{2} a^{2}+4 a^{3}+320 c_{1} \right ) \textit {\_Z}^{9}+\left (-189 \textit {\_a}^{6}+567 \textit {\_a}^{4} a -567 \textit {\_a}^{2} a^{2}+189 a^{3}+15120 c_{1} \right ) \textit {\_Z}^{6}+238140 c_{1} \textit {\_Z}^{3}+1250235 c_{1} \right )}^{3}+63}{\textit {\_a}^{2}-a}d \textit {\_a} \right )}{63}-x -c_{2} = 0 \]

✓ Solution by Mathematica

Time used: 77.065 (sec). Leaf size: 3100

DSolve[a*y[x] - y[x]^3 + y[x]*y'[x] + y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
 

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