problem 298

Internal problem ID [3554]
Internal ﬁle name [OUTPUT/3047_Sunday_June_05_2022_08_50_32_AM_54502821/index.tex]

Book: Ordinary diﬀerential equations and their solutions. By George Moseley Murphy. 1960
Section: Various 11
Problem number: 298.
ODE order: 1.
ODE degree: 1.

\[ \boxed {\left (-x^{2}+1\right ) y^{\prime }-n \left (1-2 x y+y^{2}\right )=0} \]

11.7.1 Solving as riccati ode

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

This is a Riccati ODE. Comparing the ODE to solve \[ y' = \frac {2 n x y}{x^{2}-1}-\frac {n \,y^{2}}{x^{2}-1}-\frac {n}{x^{2}-1} \] With Riccati ODE standard form \[ y' = f_0(x)+ f_1(x)y+f_2(x)y^{2} \] Shows that \(f_0(x)=-\frac {n}{x^{2}-1}\), \(f_1(x)=\frac {2 n x}{x^{2}-1}\) and \(f_2(x)=-\frac {n}{x^{2}-1}\). Let \begin {align*} y &= \frac {-u'}{f_2 u} \\ &= \frac {-u'}{-\frac {n u}{x^{2}-1}} \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 {2 n x}{\left (x^{2}-1\right )^{2}}\\ f_1 f_2 &=-\frac {2 n^{2} x}{\left (x^{2}-1\right )^{2}}\\ f_2^2 f_0 &=-\frac {n^{3}}{\left (x^{2}-1\right )^{3}} \end {align*}

Substituting the above terms back in equation (2) gives \begin {align*} -\frac {n u^{\prime \prime }\left (x \right )}{x^{2}-1}-\left (\frac {2 n x}{\left (x^{2}-1\right )^{2}}-\frac {2 n^{2} x}{\left (x^{2}-1\right )^{2}}\right ) u^{\prime }\left (x \right )-\frac {n^{3} u \left (x \right )}{\left (x^{2}-1\right )^{3}} &=0 \end {align*}

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

\[ y = \frac {\operatorname {LegendreP}\left (n , x\right ) c_{3} +\operatorname {LegendreQ}\left (n , x\right )}{\operatorname {LegendreP}\left (n -1, x\right ) c_{3} +\operatorname {LegendreQ}\left (n -1, x\right )} \]

Summary

The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {\operatorname {LegendreP}\left (n , x\right ) c_{3} +\operatorname {LegendreQ}\left (n , x\right )}{\operatorname {LegendreP}\left (n -1, x\right ) c_{3} +\operatorname {LegendreQ}\left (n -1, x\right )} \\ \end{align*}

Veriﬁcation of solutions

11.7.2 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left (-x^{2}+1\right ) y^{\prime }-n \left (1-2 x y+y^{2}\right )=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {n \left (1-2 x y+y^{2}\right )}{-x^{2}+1} \end {array} \]

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: 
   <- Abel AIR successful: ODE belongs to the 2F1 3-parameter class`

\[ y \left (x \right ) = \frac {\left (\frac {x +1}{x -1}\right )^{n} \left (-\frac {x}{2}-\frac {1}{2}\right )^{2 n} \left (16 \left (x +1\right )^{2} \left (\left (x -\frac {1}{2}\right ) n +\frac {1}{2}-\frac {x}{2}\right ) \operatorname {hypergeom}\left (\left [-n +1, -n +1\right ], \left [2-2 n \right ], -\frac {2}{x -1}\right ) c_{1} \left (\frac {x +1}{x -1}\right )^{-n}+\left (x -1\right ) \left (\left (x +1\right )^{2} n \left (\frac {x +1}{x -1}\right )^{n} \left (-\frac {x}{2}-\frac {1}{2}\right )^{-2 n} \operatorname {hypergeom}\left (\left [n , n\right ], \left [2 n \right ], -\frac {2}{x -1}\right )-16 \left (\frac {\operatorname {HeunCPrime}\left (0, 2 n -1, 0, 0, n^{2}-n +\frac {1}{2}, \frac {2}{x +1}\right ) \left (x +1\right ) \left (-\frac {x}{2}-\frac {1}{2}\right )^{-2 n}}{8}+\operatorname {HeunCPrime}\left (0, -2 n +1, 0, 0, n^{2}-n +\frac {1}{2}, \frac {2}{x +1}\right ) c_{1} \right ) \left (x -1\right )\right )\right )}{\left (x +1\right )^{2} \left (8 c_{1} \operatorname {hypergeom}\left (\left [-n +1, -n +1\right ], \left [2-2 n \right ], -\frac {2}{x -1}\right ) \left (-\frac {x}{2}-\frac {1}{2}\right )^{2 n}+\left (\frac {x +1}{x -1}\right )^{2 n} \operatorname {hypergeom}\left (\left [n , n\right ], \left [2 n \right ], -\frac {2}{x -1}\right ) \left (x -1\right )\right ) n} \]

\begin{align*} y(x)\to \frac {Q_n(x)+c_1 P_n(x)}{Q_{n-1}(x)+c_1 P_{n-1}(x)} \\ y(x)\to \frac {P_n(x)}{P_{n-1}(x)} \\ \end{align*}

11.7 problem 298

11.7.1 Solving as riccati ode

11.7.2 Maple step by step solution