Internal problem ID [10130]
Internal file name [OUTPUT/9077_Monday_June_06_2022_06_23_17_AM_11796080/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1809 (book 6.218).
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], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]
\[ \boxed {\left (y^{2}-1\right ) \left (a^{2} y^{2}-1\right ) y^{\prime \prime }+b \sqrt {\left (1-y^{2}\right ) \left (1-a^{2} y^{2}\right )}\, {y^{\prime }}^{2}+\left (1+a^{2}-2 a^{2} y^{2}\right ) y {y^{\prime }}^{2}=0} \]
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*} \left (a^{2} y^{4}-a^{2} y^{2}-y^{2}+1\right ) p \left (y \right ) \left (\frac {d}{d y}p \left (y \right )\right )+\left (b \sqrt {a^{2} y^{4}-a^{2} y^{2}-y^{2}+1}\, p \left (y \right )+p \left (y \right ) y +a^{2} p \left (y \right ) y -2 a^{2} y^{3} p \left (y \right )\right ) p \left (y \right ) = 0 \end {align*}
Which is now solved as first order ode for \(p(y)\). In canonical form the ODE is \begin {align*} p' &= F(y,p)\\ &= f( y) g(p)\\ &= -\frac {p \left (-2 a^{2} y^{3}+a^{2} y +\sqrt {\left (y -1\right ) \left (y +1\right ) \left (y a -1\right ) \left (y a +1\right )}\, b +y \right )}{a^{2} y^{4}-a^{2} y^{2}-y^{2}+1} \end {align*}
Where \(f(y)=-\frac {-2 a^{2} y^{3}+a^{2} y +\sqrt {\left (y -1\right ) \left (y +1\right ) \left (y a -1\right ) \left (y a +1\right )}\, b +y}{a^{2} y^{4}-a^{2} y^{2}-y^{2}+1}\) and \(g(p)=p\). Integrating both sides gives \begin {align*} \frac {1}{p} \,dp &= -\frac {-2 a^{2} y^{3}+a^{2} y +\sqrt {\left (y -1\right ) \left (y +1\right ) \left (y a -1\right ) \left (y a +1\right )}\, b +y}{a^{2} y^{4}-a^{2} y^{2}-y^{2}+1} \,d y\\ \int { \frac {1}{p} \,dp} &= \int {-\frac {-2 a^{2} y^{3}+a^{2} y +\sqrt {\left (y -1\right ) \left (y +1\right ) \left (y a -1\right ) \left (y a +1\right )}\, b +y}{a^{2} y^{4}-a^{2} y^{2}-y^{2}+1} \,d y}\\ \ln \left (p \right )&=\frac {\ln \left (\left (y a -1\right ) \left (a \,y^{3}-y a +y^{2}-1\right )\right )}{2}-\frac {b \,a^{2} \sqrt {-y^{2}+1}\, \sqrt {-a^{2} y^{2}+1}\, \operatorname {EllipticF}\left (y , a\right )}{\left (-1+a \right ) \left (1+a \right ) \sqrt {a^{2} y^{4}-a^{2} y^{2}-y^{2}+1}}+\frac {b \sqrt {-y^{2}+1}\, \sqrt {-a^{2} y^{2}+1}\, \operatorname {EllipticF}\left (y , a\right )}{\left (-1+a \right ) \left (1+a \right ) \sqrt {a^{2} y^{4}-a^{2} y^{2}-y^{2}+1}}+c_{1}\\ p&={\mathrm e}^{\frac {\ln \left (\left (y a -1\right ) \left (a \,y^{3}-y a +y^{2}-1\right )\right )}{2}-\frac {b \,a^{2} \sqrt {-y^{2}+1}\, \sqrt {-a^{2} y^{2}+1}\, \operatorname {EllipticF}\left (y , a\right )}{\left (-1+a \right ) \left (1+a \right ) \sqrt {a^{2} y^{4}-a^{2} y^{2}-y^{2}+1}}+\frac {b \sqrt {-y^{2}+1}\, \sqrt {-a^{2} y^{2}+1}\, \operatorname {EllipticF}\left (y , a\right )}{\left (-1+a \right ) \left (1+a \right ) \sqrt {a^{2} y^{4}-a^{2} y^{2}-y^{2}+1}}+c_{1}}\\ &=c_{1} {\mathrm e}^{\frac {\ln \left (\left (y a -1\right ) \left (a \,y^{3}-y a +y^{2}-1\right )\right )}{2}-\frac {b \,a^{2} \sqrt {-y^{2}+1}\, \sqrt {-a^{2} y^{2}+1}\, \operatorname {EllipticF}\left (y , a\right )}{\left (-1+a \right ) \left (1+a \right ) \sqrt {a^{2} y^{4}-a^{2} y^{2}-y^{2}+1}}+\frac {b \sqrt {-y^{2}+1}\, \sqrt {-a^{2} y^{2}+1}\, \operatorname {EllipticF}\left (y , a\right )}{\left (-1+a \right ) \left (1+a \right ) \sqrt {a^{2} y^{4}-a^{2} y^{2}-y^{2}+1}}} \end {align*}
Which simplifies to \[ p \left (y \right ) = c_{1} \sqrt {a^{2} y^{4}-a^{2} y^{2}-y^{2}+1}\, {\mathrm e}^{-\frac {b \,a^{2} \sqrt {-y^{2}+1}\, \sqrt {-a^{2} y^{2}+1}\, \operatorname {EllipticF}\left (y , a\right )}{\left (-1+a \right ) \left (1+a \right ) \sqrt {a^{2} y^{4}-a^{2} y^{2}-y^{2}+1}}} {\mathrm e}^{\frac {b \sqrt {-y^{2}+1}\, \sqrt {-a^{2} y^{2}+1}\, \operatorname {EllipticF}\left (y , a\right )}{\left (-1+a \right ) \left (1+a \right ) \sqrt {a^{2} y^{4}-a^{2} y^{2}-y^{2}+1}}} \] For solution (1) found earlier, since \(p=y^{\prime }\) then we now have a new first order ode to solve which is \begin {align*} y^{\prime } = c_{1} \sqrt {a^{2} y^{4}-y^{2}-a^{2} y^{2}+1}\, {\mathrm e}^{-\frac {b \,a^{2} \sqrt {1-y^{2}}\, \sqrt {1-a^{2} y^{2}}\, \operatorname {EllipticF}\left (y, a\right )}{\left (-1+a \right ) \left (1+a \right ) \sqrt {a^{2} y^{4}-y^{2}-a^{2} y^{2}+1}}} {\mathrm e}^{\frac {b \sqrt {1-y^{2}}\, \sqrt {1-a^{2} y^{2}}\, \operatorname {EllipticF}\left (y, a\right )}{\left (-1+a \right ) \left (1+a \right ) \sqrt {a^{2} y^{4}-y^{2}-a^{2} y^{2}+1}}} \end {align*}
Integrating both sides gives \begin{align*} \int \frac {{\mathrm e}^{\frac {b \,a^{2} \sqrt {-y^{2}+1}\, \sqrt {-a^{2} y^{2}+1}\, \operatorname {EllipticF}\left (y , a\right )}{\left (-1+a \right ) \left (1+a \right ) \sqrt {a^{2} y^{4}-a^{2} y^{2}-y^{2}+1}}} {\mathrm e}^{-\frac {b \sqrt {-y^{2}+1}\, \sqrt {-a^{2} y^{2}+1}\, \operatorname {EllipticF}\left (y , a\right )}{\left (-1+a \right ) \left (1+a \right ) \sqrt {a^{2} y^{4}-a^{2} y^{2}-y^{2}+1}}}}{c_{1} \sqrt {a^{2} y^{4}-a^{2} y^{2}-y^{2}+1}}d y &= \int d x \\ \frac {{\mathrm e}^{\frac {b \sqrt {1-y^{2}}\, \sqrt {1-a^{2} y^{2}}\, \operatorname {EllipticF}\left (y, a\right )}{\sqrt {a^{2} y^{4}-y^{2}-a^{2} y^{2}+1}}}}{c_{1} b}&=x +c_{2} \\ \end{align*}
The solution(s) found are the following \begin{align*} \tag{1} \frac {{\mathrm e}^{\frac {b \sqrt {1-y^{2}}\, \sqrt {1-a^{2} y^{2}}\, \operatorname {EllipticF}\left (y, a\right )}{\sqrt {a^{2} y^{4}-y^{2}-a^{2} y^{2}+1}}}}{c_{1} b} &= x +c_{2} \\ \end{align*}
Verification of solutions
\[ \frac {{\mathrm e}^{\frac {b \sqrt {1-y^{2}}\, \sqrt {1-a^{2} y^{2}}\, \operatorname {EllipticF}\left (y, a\right )}{\sqrt {a^{2} y^{4}-y^{2}-a^{2} y^{2}+1}}}}{c_{1} b} = x +c_{2} \] Verified OK.
Maple trace
`Methods for second order ODEs: --- Trying classification methods --- trying 2nd order Liouville <- 2nd_order Liouville successful`
✗ Solution by Maple
dsolve((y(x)^2-1)*(a^2*y(x)^2-1)*diff(diff(y(x),x),x)+b*((1-y(x)^2)*(1-a^2*y(x)^2))^(1/2)*diff(y(x),x)^2+(1+a^2-2*a^2*y(x)^2)*y(x)*diff(y(x),x)^2=0,y(x), singsol=all)
\[ \text {No solution found} \]
✓ Solution by Mathematica
Time used: 22.452 (sec). Leaf size: 372
DSolve[y[x]*(1 + a^2 - 2*a^2*y[x]^2)*y'[x]^2 + b*Sqrt[(1 - y[x]^2)*(1 - a^2*y[x]^2)]*y'[x]^2 + (-1 + y[x]^2)*(-1 + a^2*y[x]^2)*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\exp \left (\frac {b \sqrt {1-K[1]^2} \sqrt {1-a^2 K[1]^2} \operatorname {EllipticF}\left (\arcsin (K[1]),a^2\right )}{\sqrt {\left (K[1]^2-1\right ) \left (a^2 K[1]^2-1\right )}}+\frac {1}{2} (-\log (1-K[1])-\log (K[1]+1)-\log (1-a K[1])-\log (a K[1]+1))\right )}{c_1}dK[1]\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}-\frac {\exp \left (\frac {b \sqrt {1-K[1]^2} \sqrt {1-a^2 K[1]^2} \operatorname {EllipticF}\left (\arcsin (K[1]),a^2\right )}{\sqrt {\left (K[1]^2-1\right ) \left (a^2 K[1]^2-1\right )}}+\frac {1}{2} (-\log (1-K[1])-\log (K[1]+1)-\log (1-a K[1])-\log (a K[1]+1))\right )}{c_1}dK[1]\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\exp \left (\frac {b \sqrt {1-K[1]^2} \sqrt {1-a^2 K[1]^2} \operatorname {EllipticF}\left (\arcsin (K[1]),a^2\right )}{\sqrt {\left (K[1]^2-1\right ) \left (a^2 K[1]^2-1\right )}}+\frac {1}{2} (-\log (1-K[1])-\log (K[1]+1)-\log (1-a K[1])-\log (a K[1]+1))\right )}{c_1}dK[1]\&\right ][x+c_2] \\ \end{align*}