problem 1837 (book 6.246)

Internal problem ID [10158]
Internal ﬁle name [OUTPUT/9105_Monday_June_06_2022_06_39_18_AM_16357301/index.tex]

Book: Diﬀerential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1837 (book 6.246).
ODE order: 2.
ODE degree: 2.

\[ \boxed {\sqrt {a {y^{\prime \prime }}^{2}+{y^{\prime }}^{2} b}+c y y^{\prime \prime }+d {y^{\prime }}^{2}=0} \]

7.246.1 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*} c y p \left (y \right ) \left (\frac {d}{d y}p \left (y \right )\right )+d p \left (y \right )^{2} = -\sqrt {a p \left (y \right )^{2} \left (\frac {d}{d y}p \left (y \right )\right )^{2}+p \left (y \right )^{2} b} \end {align*}

Which is now solved as ﬁrst order ode for \(p(y)\). Solving the given ode for \(\frac {d}{d y}p \left (y \right )\) results in \(2\) diﬀerential equations to solve. Each one of these will generate a solution. The equations generated are \begin {align*} \frac {d}{d y}p \left (y \right )&=\frac {p \left (y \right ) c d y +\sqrt {p \left (y \right )^{2} a \,d^{2}+y^{2} b \,c^{2}-a b}}{-c^{2} y^{2}+a} \tag {1} \\ \frac {d}{d y}p \left (y \right )&=\frac {p \left (y \right ) c d y -\sqrt {p \left (y \right )^{2} a \,d^{2}+y^{2} b \,c^{2}-a b}}{-c^{2} y^{2}+a} \tag {2} \end {align*}

Maple trace

`Methods for second order ODEs: 
   *** Sublevel 2 *** 
   Methods for second order ODEs: 
   Successful isolation of d^2y/dx^2: 2 solutions were found. Trying to solve each resulting ODE. 
      *** Sublevel 3 *** 
      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)-(-c*d*_a*_b(_a)+(_b(_a)^2*a*d^2+_a^2*b*c^2-a*b)^(1/2))*_b(_a)/(_a^ 
         Methods for first order ODEs: 
         --- Trying classification methods --- 
         trying homogeneous types: 
         trying exact 
         Looking for potential symmetries 
         trying an equivalence to an Abel ODE 
         trying 1st order ODE linearizable_by_differentiation 
      -> trying 2nd order, dynamical_symmetries, fully reducible to Abel through one integrating factor of the form G(x,y)/(1+H(x,y) 
      trying 2nd order, integrating factors of the form mu(x,y)/(y)^n, only the singular cases 
      trying differential order: 2; exact nonlinear 
      trying 2nd order, integrating factor of the form mu(x,y) 
      -> trying 2nd order, the S-function method 
         -> trying a change of variables {x -> y(x), y(x) -> x} and re-entering methods for the S-function 
         -> trying 2nd order, the S-function method 
         -> trying 2nd order, No Point Symmetries Class V 
         -> trying 2nd order, No Point Symmetries Class V 
         -> trying 2nd order, No Point Symmetries Class V 
      trying 2nd order, integrating factor of the form mu(x,y)/(y)^n, only the general case 
      -> trying 2nd order, dynamical_symmetries, only a reduction of order through one integrating factor of the form G(x,y)/(1+H(x, 
   solving 2nd order ODE of high degree, Lie methods 
`, `2nd order, trying reduction of order with given symmetries:`[1, 0]

7.246 problem 1837 (book 6.246)

7.246.1 Solving as second order ode missing x ode