Internal problem ID [10348]
Internal file name [OUTPUT/9296_Monday_June_06_2022_01_49_22_PM_49385801/index.tex]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev.
Second edition
Section: Chapter 1, section 1.2. Riccati Equation. 1.2.2. Equations Containing Power
Functions
Problem number: 19.
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]
\[ \boxed {a \,x^{2} \left (x -1\right )^{2} \left (y^{\prime }+\lambda y^{2}\right )=-b \,x^{2}-c x -s} \]
In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= -\frac {y^{2} a \lambda \,x^{4}-2 y^{2} a \lambda \,x^{3}+y^{2} a \lambda \,x^{2}+b \,x^{2}+c x +s}{a \,x^{2} \left (x -1\right )^{2}} \end {align*}
This is a Riccati ODE. Comparing the ODE to solve \[ y' = -\frac {x^{2} y^{2} \lambda }{\left (x -1\right )^{2}}+\frac {2 x \,y^{2} \lambda }{\left (x -1\right )^{2}}-\frac {y^{2} \lambda }{\left (x -1\right )^{2}}-\frac {b}{a \left (x -1\right )^{2}}-\frac {c}{a x \left (x -1\right )^{2}}-\frac {s}{a \,x^{2} \left (x -1\right )^{2}} \] With Riccati ODE standard form \[ y' = f_0(x)+ f_1(x)y+f_2(x)y^{2} \] Shows that \(f_0(x)=-\frac {b \,x^{2}+c x +s}{a \,x^{2} \left (x -1\right )^{2}}\), \(f_1(x)=0\) and \(f_2(x)=-\frac {a \lambda \,x^{4}-2 a \lambda \,x^{3}+a \lambda \,x^{2}}{a \,x^{2} \left (x -1\right )^{2}}\). Let \begin {align*} y &= \frac {-u'}{f_2 u} \\ &= \frac {-u'}{-\frac {\left (a \lambda \,x^{4}-2 a \lambda \,x^{3}+a \lambda \,x^{2}\right ) u}{a \,x^{2} \left (x -1\right )^{2}}} \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 {4 a \lambda \,x^{3}-6 a \lambda \,x^{2}+2 a \lambda x}{a \,x^{2} \left (x -1\right )^{2}}+\frac {2 a \lambda \,x^{4}-4 a \lambda \,x^{3}+2 a \lambda \,x^{2}}{a \,x^{3} \left (x -1\right )^{2}}+\frac {2 a \lambda \,x^{4}-4 a \lambda \,x^{3}+2 a \lambda \,x^{2}}{a \,x^{2} \left (x -1\right )^{3}}\\ f_1 f_2 &=0\\ f_2^2 f_0 &=-\frac {\left (a \lambda \,x^{4}-2 a \lambda \,x^{3}+a \lambda \,x^{2}\right )^{2} \left (b \,x^{2}+c x +s \right )}{a^{3} x^{6} \left (x -1\right )^{6}} \end {align*}
Substituting the above terms back in equation (2) gives \begin {align*} -\frac {\left (a \lambda \,x^{4}-2 a \lambda \,x^{3}+a \lambda \,x^{2}\right ) u^{\prime \prime }\left (x \right )}{a \,x^{2} \left (x -1\right )^{2}}-\left (-\frac {4 a \lambda \,x^{3}-6 a \lambda \,x^{2}+2 a \lambda x}{a \,x^{2} \left (x -1\right )^{2}}+\frac {2 a \lambda \,x^{4}-4 a \lambda \,x^{3}+2 a \lambda \,x^{2}}{a \,x^{3} \left (x -1\right )^{2}}+\frac {2 a \lambda \,x^{4}-4 a \lambda \,x^{3}+2 a \lambda \,x^{2}}{a \,x^{2} \left (x -1\right )^{3}}\right ) u^{\prime }\left (x \right )-\frac {\left (a \lambda \,x^{4}-2 a \lambda \,x^{3}+a \lambda \,x^{2}\right )^{2} \left (b \,x^{2}+c x +s \right ) u \left (x \right )}{a^{3} x^{6} \left (x -1\right )^{6}} &=0 \end {align*}
Solving the above ODE (this ode solved using Maple, not this program), gives
\[ u \left (x \right ) = \left (x -1\right )^{\frac {\sqrt {a}-\sqrt {\left (-4 s -4 c -4 b \right ) \lambda +a}}{2 \sqrt {a}}} \left (c_{2} x^{-\frac {-\sqrt {a}+\sqrt {-4 \lambda s +a}}{2 \sqrt {a}}} \operatorname {hypergeom}\left (\left [-\frac {\sqrt {\left (-4 s -4 c -4 b \right ) \lambda +a}-\sqrt {a}+\sqrt {-4 \lambda s +a}-\sqrt {-4 b \lambda +a}}{2 \sqrt {a}}, -\frac {\sqrt {\left (-4 s -4 c -4 b \right ) \lambda +a}-\sqrt {a}+\sqrt {-4 \lambda s +a}+\sqrt {-4 b \lambda +a}}{2 \sqrt {a}}\right ], \left [1-\frac {\sqrt {-4 \lambda s +a}}{\sqrt {a}}\right ], x\right )+c_{1} x^{\frac {\sqrt {-4 \lambda s +a}+\sqrt {a}}{2 \sqrt {a}}} \operatorname {hypergeom}\left (\left [\frac {-\sqrt {\left (-4 s -4 c -4 b \right ) \lambda +a}+\sqrt {a}+\sqrt {-4 \lambda s +a}+\sqrt {-4 b \lambda +a}}{2 \sqrt {a}}, \frac {-\sqrt {\left (-4 s -4 c -4 b \right ) \lambda +a}+\sqrt {a}+\sqrt {-4 \lambda s +a}-\sqrt {-4 b \lambda +a}}{2 \sqrt {a}}\right ], \left [1+\frac {\sqrt {-4 \lambda s +a}}{\sqrt {a}}\right ], x\right )\right ) \] The above shows that \[ u^{\prime }\left (x \right ) = -\frac {\left (2 s \sqrt {\left (-4 s -4 c -4 b \right ) \lambda +a}+\left (-c -2 s \right ) \sqrt {-4 \lambda s +a}+\sqrt {a}\, c \right ) x^{\frac {\sqrt {-4 \lambda s +a}+\sqrt {a}}{2 \sqrt {a}}} \left (x -1\right )^{\frac {\sqrt {a}-\sqrt {\left (-4 s -4 c -4 b \right ) \lambda +a}}{2 \sqrt {a}}} c_{1} \operatorname {hypergeom}\left (\left [\frac {-\sqrt {\left (-4 s -4 c -4 b \right ) \lambda +a}+3 \sqrt {a}+\sqrt {-4 \lambda s +a}+\sqrt {-4 b \lambda +a}}{2 \sqrt {a}}, -\frac {\sqrt {\left (-4 s -4 c -4 b \right ) \lambda +a}-3 \sqrt {a}-\sqrt {-4 \lambda s +a}+\sqrt {-4 b \lambda +a}}{2 \sqrt {a}}\right ], \left [\frac {2 \sqrt {a}+\sqrt {-4 \lambda s +a}}{\sqrt {a}}\right ], x\right )+\left (x -1\right )^{\frac {\sqrt {a}-\sqrt {\left (-4 s -4 c -4 b \right ) \lambda +a}}{2 \sqrt {a}}} \left (2 s \sqrt {\left (-4 s -4 c -4 b \right ) \lambda +a}+\left (c +2 s \right ) \sqrt {-4 \lambda s +a}+\sqrt {a}\, c \right ) x^{-\frac {-\sqrt {a}+\sqrt {-4 \lambda s +a}}{2 \sqrt {a}}} c_{2} \operatorname {hypergeom}\left (\left [-\frac {\sqrt {\left (-4 s -4 c -4 b \right ) \lambda +a}-3 \sqrt {a}+\sqrt {-4 \lambda s +a}-\sqrt {-4 b \lambda +a}}{2 \sqrt {a}}, -\frac {\sqrt {\left (-4 s -4 c -4 b \right ) \lambda +a}-3 \sqrt {a}+\sqrt {-4 \lambda s +a}+\sqrt {-4 b \lambda +a}}{2 \sqrt {a}}\right ], \left [\frac {2 \sqrt {a}-\sqrt {-4 \lambda s +a}}{\sqrt {a}}\right ], x\right )-2 \left (\left (x^{-\frac {\sqrt {-4 \lambda s +a}+\sqrt {a}}{2 \sqrt {a}}} \left (\sqrt {a}-\sqrt {-4 \lambda s +a}\right ) \left (x -1\right )^{\frac {\sqrt {a}-\sqrt {\left (-4 s -4 c -4 b \right ) \lambda +a}}{2 \sqrt {a}}}+x^{-\frac {-\sqrt {a}+\sqrt {-4 \lambda s +a}}{2 \sqrt {a}}} \left (x -1\right )^{-\frac {\sqrt {a}+\sqrt {\left (-4 s -4 c -4 b \right ) \lambda +a}}{2 \sqrt {a}}} \left (\sqrt {a}-\sqrt {\left (-4 s -4 c -4 b \right ) \lambda +a}\right )\right ) c_{2} \operatorname {hypergeom}\left (\left [-\frac {\sqrt {\left (-4 s -4 c -4 b \right ) \lambda +a}-\sqrt {a}+\sqrt {-4 \lambda s +a}-\sqrt {-4 b \lambda +a}}{2 \sqrt {a}}, -\frac {\sqrt {\left (-4 s -4 c -4 b \right ) \lambda +a}-\sqrt {a}+\sqrt {-4 \lambda s +a}+\sqrt {-4 b \lambda +a}}{2 \sqrt {a}}\right ], \left [\frac {\sqrt {a}-\sqrt {-4 \lambda s +a}}{\sqrt {a}}\right ], x\right )+c_{1} \operatorname {hypergeom}\left (\left [\frac {-\sqrt {\left (-4 s -4 c -4 b \right ) \lambda +a}+\sqrt {a}+\sqrt {-4 \lambda s +a}+\sqrt {-4 b \lambda +a}}{2 \sqrt {a}}, \frac {-\sqrt {\left (-4 s -4 c -4 b \right ) \lambda +a}+\sqrt {a}+\sqrt {-4 \lambda s +a}-\sqrt {-4 b \lambda +a}}{2 \sqrt {a}}\right ], \left [\frac {\sqrt {-4 \lambda s +a}+\sqrt {a}}{\sqrt {a}}\right ], x\right ) \left (x^{\frac {-\sqrt {a}+\sqrt {-4 \lambda s +a}}{2 \sqrt {a}}} \left (\sqrt {-4 \lambda s +a}+\sqrt {a}\right ) \left (x -1\right )^{\frac {\sqrt {a}-\sqrt {\left (-4 s -4 c -4 b \right ) \lambda +a}}{2 \sqrt {a}}}+x^{\frac {\sqrt {-4 \lambda s +a}+\sqrt {a}}{2 \sqrt {a}}} \left (x -1\right )^{-\frac {\sqrt {a}+\sqrt {\left (-4 s -4 c -4 b \right ) \lambda +a}}{2 \sqrt {a}}} \left (\sqrt {a}-\sqrt {\left (-4 s -4 c -4 b \right ) \lambda +a}\right )\right )\right ) s}{4 \sqrt {a}\, s} \] Using the above in (1) gives the solution \[ \text {Expression too large to display} \] Dividing both numerator and denominator by \(c_{1}\) gives, after renaming the constant \(\frac {c_{2}}{c_{1}}=c_{3}\) the following solution
\[ \text {Expression too large to display} \]
The solution(s) found are the following \begin{align*} \tag{1} \text {Expression too large to display} \\ \end{align*}
Verification of solutions
\[ \text {Expression too large to display} \] Warning, solution could not be verified
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & a \,x^{2} \left (x -1\right )^{2} \left (y^{\prime }+\lambda y^{2}\right )=-b \,x^{2}-c x -s \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-\frac {y^{2} a \lambda \,x^{4}-2 y^{2} a \lambda \,x^{3}+y^{2} a \lambda \,x^{2}+b \,x^{2}+c x +s}{a \,x^{2} \left (x -1\right )^{2}} \end {array} \]
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) = -(b*x^2+c*x+s)*lambda*y(x)/(a*x^2*(x^2-2*x+1)), y(x)` *** Subleve 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 -> Legendre -> Whittaker -> hyper3: Equivalence to 1F1 under a power @ Moebius -> hypergeometric -> heuristic approach <- heuristic approach successful <- hypergeometric successful <- special function solution successful <- Riccati to 2nd Order successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 1087
dsolve(a*x^2*(x-1)^2*(diff(y(x),x)+lambda*y(x)^2)+b*x^2+c*x+s=0,y(x), singsol=all)
\[ \text {Expression too large to display} \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[a*x^2*(x-1)^2*(y'[x]+\[Lambda]*y[x]^2)+b*x^2+c*x+s==0,y[x],x,IncludeSingularSolutions -> True]
Timed out