problem 363

Internal problem ID [3619]
Internal ﬁle name [OUTPUT/3112_Sunday_June_05_2022_08_52_28_AM_80988955/index.tex]

Book: Ordinary diﬀerential equations and their solutions. By George Moseley Murphy. 1960
Section: Various 13
Problem number: 363.
ODE order: 1.
ODE degree: 1.

\[ \boxed {x \left (-x^{2}+1\right ) y^{\prime }+\left (-x^{2}+1\right ) y^{2}=-x^{2}} \]

13.9.1 Solving as riccati ode

In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= -\frac {x^{2} y^{2}-x^{2}-y^{2}}{x \left (x^{2}-1\right )} \end {align*}

This is a Riccati ODE. Comparing the ODE to solve \[ y' = -\frac {x \,y^{2}}{x^{2}-1}+\frac {x}{x^{2}-1}+\frac {y^{2}}{x \left (x^{2}-1\right )} \] With Riccati ODE standard form \[ y' = f_0(x)+ f_1(x)y+f_2(x)y^{2} \] Shows that \(f_0(x)=\frac {x}{x^{2}-1}\), \(f_1(x)=0\) and \(f_2(x)=-\frac {1}{x}\). Let \begin {align*} y &= \frac {-u'}{f_2 u} \\ &= \frac {-u'}{-\frac {u}{x}} \tag {1} \end {align*}

Using the above substitution in the given ODE results (after some simpliﬁcation)in a second order ODE to solve for \(u(x)\) which is \begin {align*} f_2 u''(x) -\left ( f_2' + f_1 f_2 \right ) u'(x) + f_2^2 f_0 u(x) &= 0 \tag {2} \end {align*}

But \begin {align*} f_2' &=\frac {1}{x^{2}}\\ f_1 f_2 &=0\\ f_2^2 f_0 &=\frac {1}{x \left (x^{2}-1\right )} \end {align*}

Substituting the above terms back in equation (2) gives \begin {align*} -\frac {u^{\prime \prime }\left (x \right )}{x}-\frac {u^{\prime }\left (x \right )}{x^{2}}+\frac {u \left (x \right )}{x \left (x^{2}-1\right )} &=0 \end {align*}

\[ u \left (x \right ) = c_{1} \operatorname {EllipticE}\left (x \right )+c_{2} \left (\operatorname {EllipticCE}\left (x \right )-\operatorname {EllipticCK}\left (x \right )\right ) \] The above shows that \[ u^{\prime }\left (x \right ) = \frac {c_{2} \operatorname {EllipticCE}\left (x \right )+c_{1} \left (\operatorname {EllipticE}\left (x \right )-\operatorname {EllipticK}\left (x \right )\right )}{x} \] Using the above in (1) gives the solution \[ y = \frac {c_{2} \operatorname {EllipticCE}\left (x \right )+c_{1} \left (\operatorname {EllipticE}\left (x \right )-\operatorname {EllipticK}\left (x \right )\right )}{c_{1} \operatorname {EllipticE}\left (x \right )+c_{2} \left (\operatorname {EllipticCE}\left (x \right )-\operatorname {EllipticCK}\left (x \right )\right )} \] Dividing both numerator and denominator by \(c_{1}\) gives, after renaming the constant \(\frac {c_{2}}{c_{1}}=c_{3}\) the following solution

\[ y = \frac {\operatorname {EllipticCE}\left (x \right )+c_{3} \left (\operatorname {EllipticE}\left (x \right )-\operatorname {EllipticK}\left (x \right )\right )}{c_{3} \operatorname {EllipticE}\left (x \right )+\operatorname {EllipticCE}\left (x \right )-\operatorname {EllipticCK}\left (x \right )} \]

Summary

The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {\operatorname {EllipticCE}\left (x \right )+c_{3} \left (\operatorname {EllipticE}\left (x \right )-\operatorname {EllipticK}\left (x \right )\right )}{c_{3} \operatorname {EllipticE}\left (x \right )+\operatorname {EllipticCE}\left (x \right )-\operatorname {EllipticCK}\left (x \right )} \\ \end{align*}

Veriﬁcation of solutions

13.9.2 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x \left (-x^{2}+1\right ) y^{\prime }+\left (-x^{2}+1\right ) y^{2}=-x^{2} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-\frac {x^{2}+\left (-x^{2}+1\right ) y^{2}}{x \left (-x^{2}+1\right )} \end {array} \]

Maple trace

`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 
Looking for potential symmetries 
trying Riccati 
trying Riccati sub-methods: 
   trying Riccati to 2nd Order 
   -> Calling odsolve with the ODE`, diff(diff(y(x), x), x) = -(diff(y(x), x))/x+y(x)/(x^2-1), y(x)`      *** Sublevel 2 *** 
      Methods for second order ODEs: 
      --- Trying classification methods --- 
      trying a quadrature 
      checking if the LODE has constant coefficients 
      checking if the LODE is of Euler type 
      trying a symmetry of the form [xi=0, eta=F(x)] 
      checking if the LODE is missing y 
      -> Trying a Liouvillian solution using Kovacics algorithm 
      <- No Liouvillian solutions exists 
      -> Trying a solution in terms of special functions: 
         -> Bessel 
         -> elliptic 
         <- elliptic successful 
      <- special function solution successful 
   <- Riccati to 2nd Order successful`

\[ y \left (x \right ) = \frac {c_{1} \operatorname {EllipticCE}\left (x \right )+\operatorname {EllipticE}\left (x \right )-\operatorname {EllipticK}\left (x \right )}{c_{1} \operatorname {EllipticCE}\left (x \right )-c_{1} \operatorname {EllipticCK}\left (x \right )+\operatorname {EllipticE}\left (x \right )} \]

\begin{align*} y(x)\to -\frac {2 \left (\pi G_{2,2}^{2,0}\left (x^2| \begin {array}{c} \frac {1}{2},\frac {3}{2} \\ 0,1 \\ \end {array} \right )+c_1 \left (\operatorname {EllipticK}\left (x^2\right )-\operatorname {EllipticE}\left (x^2\right )\right )\right )}{\pi G_{2,2}^{2,0}\left (x^2| \begin {array}{c} \frac {1}{2},\frac {3}{2} \\ 0,0 \\ \end {array} \right )+2 c_1 \operatorname {EllipticE}\left (x^2\right )} \\ y(x)\to 1-\frac {\operatorname {EllipticK}\left (x^2\right )}{\operatorname {EllipticE}\left (x^2\right )} \\ \end{align*}

13.9 problem 363

13.9.1 Solving as riccati ode

13.9.2 Maple step by step solution