Internal problem ID [8514]
Internal file name [OUTPUT/7447_Sunday_June_05_2022_10_55_19_PM_24590031/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 178.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "riccati"
Maple gives the following as the ode type
[_rational, _Riccati]
\[ \boxed {2 x \left (x^{2}-1\right ) y^{\prime }+2 \left (x^{2}-1\right ) y^{2}-\left (3 x^{2}-5\right ) y=-x^{2}+3} \]
In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= -\frac {2 y^{2} x^{2}-3 x^{2} y +x^{2}-2 y^{2}+5 y -3}{2 x \left (x^{2}-1\right )} \end {align*}
This is a Riccati ODE. Comparing the ODE to solve \[ y' = -\frac {x \,y^{2}}{x^{2}-1}+\frac {3 x y}{2 \left (x^{2}-1\right )}-\frac {x}{2 \left (x^{2}-1\right )}+\frac {y^{2}}{x \left (x^{2}-1\right )}-\frac {5 y}{2 x \left (x^{2}-1\right )}+\frac {3}{2 x \left (x^{2}-1\right )} \] With Riccati ODE standard form \[ y' = f_0(x)+ f_1(x)y+f_2(x)y^{2} \] Shows that \(f_0(x)=-\frac {x^{2}-3}{2 x \left (x^{2}-1\right )}\), \(f_1(x)=-\frac {-3 x^{2}+5}{2 x \left (x^{2}-1\right )}\) and \(f_2(x)=-\frac {2 x^{2}-2}{2 x \left (x^{2}-1\right )}\). Let \begin {align*} y &= \frac {-u'}{f_2 u} \\ &= \frac {-u'}{-\frac {\left (2 x^{2}-2\right ) u}{2 x \left (x^{2}-1\right )}} \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}{x^{2}-1}+\frac {2 x^{2}-2}{2 x^{2} \left (x^{2}-1\right )}+\frac {2 x^{2}-2}{\left (x^{2}-1\right )^{2}}\\ f_1 f_2 &=\frac {\left (-3 x^{2}+5\right ) \left (2 x^{2}-2\right )}{4 x^{2} \left (x^{2}-1\right )^{2}}\\ f_2^2 f_0 &=-\frac {\left (2 x^{2}-2\right )^{2} \left (x^{2}-3\right )}{8 x^{3} \left (x^{2}-1\right )^{3}} \end {align*}
Substituting the above terms back in equation (2) gives \begin {align*} -\frac {\left (2 x^{2}-2\right ) u^{\prime \prime }\left (x \right )}{2 x \left (x^{2}-1\right )}-\left (-\frac {2}{x^{2}-1}+\frac {2 x^{2}-2}{2 x^{2} \left (x^{2}-1\right )}+\frac {2 x^{2}-2}{\left (x^{2}-1\right )^{2}}+\frac {\left (-3 x^{2}+5\right ) \left (2 x^{2}-2\right )}{4 x^{2} \left (x^{2}-1\right )^{2}}\right ) u^{\prime }\left (x \right )-\frac {\left (2 x^{2}-2\right )^{2} \left (x^{2}-3\right ) u \left (x \right )}{8 x^{3} \left (x^{2}-1\right )^{3}} &=0 \end {align*}
Solving the above ODE (this ode solved using Maple, not this program), gives
\[ u \left (x \right ) = x c_{1} +c_{2} \operatorname {LegendreQ}\left (-\frac {1}{4}, \frac {1}{4}, \sqrt {-x^{2}+1}\right ) x^{\frac {5}{4}} \] The above shows that \[ u^{\prime }\left (x \right ) = \frac {c_{2} \operatorname {LegendreQ}\left (-\frac {1}{4}, \frac {1}{4}, \sqrt {-x^{2}+1}\right ) x^{\frac {1}{4}} \sqrt {-x^{2}+1}+x^{\frac {1}{4}} \operatorname {LegendreQ}\left (\frac {3}{4}, \frac {1}{4}, \sqrt {-x^{2}+1}\right ) c_{2} +2 c_{1} \sqrt {-x^{2}+1}}{2 \sqrt {-x^{2}+1}} \] Using the above in (1) gives the solution \[ y = \frac {\left (c_{2} \operatorname {LegendreQ}\left (-\frac {1}{4}, \frac {1}{4}, \sqrt {-x^{2}+1}\right ) x^{\frac {1}{4}} \sqrt {-x^{2}+1}+x^{\frac {1}{4}} \operatorname {LegendreQ}\left (\frac {3}{4}, \frac {1}{4}, \sqrt {-x^{2}+1}\right ) c_{2} +2 c_{1} \sqrt {-x^{2}+1}\right ) x \left (x^{2}-1\right )}{\sqrt {-x^{2}+1}\, \left (2 x^{2}-2\right ) \left (x c_{1} +c_{2} \operatorname {LegendreQ}\left (-\frac {1}{4}, \frac {1}{4}, \sqrt {-x^{2}+1}\right ) x^{\frac {5}{4}}\right )} \] Dividing both numerator and denominator by \(c_{1}\) gives, after renaming the constant \(\frac {c_{2}}{c_{1}}=c_{3}\) the following solution
\[ y = \frac {\operatorname {LegendreQ}\left (-\frac {1}{4}, \frac {1}{4}, \sqrt {-x^{2}+1}\right ) x^{\frac {1}{4}} \sqrt {-x^{2}+1}+x^{\frac {1}{4}} \operatorname {LegendreQ}\left (\frac {3}{4}, \frac {1}{4}, \sqrt {-x^{2}+1}\right )+2 c_{3} \sqrt {-x^{2}+1}}{\sqrt {-x^{2}+1}\, \left (2 \operatorname {LegendreQ}\left (-\frac {1}{4}, \frac {1}{4}, \sqrt {-x^{2}+1}\right ) x^{\frac {1}{4}}+2 c_{3} \right )} \]
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {\operatorname {LegendreQ}\left (-\frac {1}{4}, \frac {1}{4}, \sqrt {-x^{2}+1}\right ) x^{\frac {1}{4}} \sqrt {-x^{2}+1}+x^{\frac {1}{4}} \operatorname {LegendreQ}\left (\frac {3}{4}, \frac {1}{4}, \sqrt {-x^{2}+1}\right )+2 c_{3} \sqrt {-x^{2}+1}}{\sqrt {-x^{2}+1}\, \left (2 \operatorname {LegendreQ}\left (-\frac {1}{4}, \frac {1}{4}, \sqrt {-x^{2}+1}\right ) x^{\frac {1}{4}}+2 c_{3} \right )} \\ \end{align*}
Verification of solutions
\[ y = \frac {\operatorname {LegendreQ}\left (-\frac {1}{4}, \frac {1}{4}, \sqrt {-x^{2}+1}\right ) x^{\frac {1}{4}} \sqrt {-x^{2}+1}+x^{\frac {1}{4}} \operatorname {LegendreQ}\left (\frac {3}{4}, \frac {1}{4}, \sqrt {-x^{2}+1}\right )+2 c_{3} \sqrt {-x^{2}+1}}{\sqrt {-x^{2}+1}\, \left (2 \operatorname {LegendreQ}\left (-\frac {1}{4}, \frac {1}{4}, \sqrt {-x^{2}+1}\right ) x^{\frac {1}{4}}+2 c_{3} \right )} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & 2 x \left (x^{2}-1\right ) y^{\prime }+2 \left (x^{2}-1\right ) y^{2}-\left (3 x^{2}-5\right ) y=-x^{2}+3 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {-2 \left (x^{2}-1\right ) y^{2}+\left (3 x^{2}-5\right ) y-x^{2}+3}{2 x \left (x^{2}-1\right )} \end {array} \]
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: <- Riccati particular case Kamke (a) successful`
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 105
dsolve(2*x*(x^2-1)*diff(y(x),x) + 2*(x^2-1)*y(x)^2 - (3*x^2-5)*y(x) + x^2 - 3=0,y(x), singsol=all)
\[ y \left (x \right ) = \frac {2 \sqrt {2}\, \operatorname {EllipticF}\left (\sqrt {x +1}, \frac {\sqrt {2}}{2}\right ) \sqrt {-x}\, \sqrt {1-x}\, \sqrt {x +1}-\sqrt {x -1}\, \sqrt {x}\, \sqrt {x +1}\, c_{1} +2 x}{\sqrt {x +1}\, \left (2 \operatorname {EllipticF}\left (\sqrt {x +1}, \frac {\sqrt {2}}{2}\right ) \sqrt {-x}\, \sqrt {2}\, \sqrt {1-x}-c_{1} \sqrt {x}\, \sqrt {x -1}\right )} \]
✓ Solution by Mathematica
Time used: 20.302 (sec). Leaf size: 54
DSolve[2*x*(x^2-1)*y'[x] + 2*(x^2-1)*y[x]^2 - (3*x^2-5)*y[x] + x^2 - 3==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to 1+\frac {\sqrt {x}}{\sqrt {1-x^2} \left (2 \sqrt {x} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},x^2\right )+c_1\right )} \\ y(x)\to 1 \\ \end{align*}