Internal problem ID [10400]
Internal file name [OUTPUT/9348_Monday_June_06_2022_02_14_54_PM_33630119/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: 71.
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]
\[ \boxed {a \left (x^{2}-1\right ) \left (y^{\prime }+\lambda y^{2}\right )+b x \left (x^{2}-1\right ) y=-c \,x^{2}-d x -s} \]
In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= -\frac {y^{2} a \lambda \,x^{2}+b \,x^{3} y -y^{2} a \lambda -b x y +c \,x^{2}+d x +s}{a \left (x^{2}-1\right )} \end {align*}
This is a Riccati ODE. Comparing the ODE to solve \[ y' = -\frac {y^{2} \lambda \,x^{2}}{x^{2}-1}-\frac {b \,x^{3} y}{a \left (x^{2}-1\right )}+\frac {y^{2} \lambda }{x^{2}-1}+\frac {b x y}{a \left (x^{2}-1\right )}-\frac {c \,x^{2}}{a \left (x^{2}-1\right )}-\frac {d x}{a \left (x^{2}-1\right )}-\frac {s}{a \left (x^{2}-1\right )} \] With Riccati ODE standard form \[ y' = f_0(x)+ f_1(x)y+f_2(x)y^{2} \] Shows that \(f_0(x)=-\frac {c \,x^{2}+d x +s}{a \left (x^{2}-1\right )}\), \(f_1(x)=-\frac {b \,x^{3}-b x}{a \left (x^{2}-1\right )}\) and \(f_2(x)=-\frac {\lambda \,x^{2} a -a \lambda }{a \left (x^{2}-1\right )}\). Let \begin {align*} y &= \frac {-u'}{f_2 u} \\ &= \frac {-u'}{-\frac {\left (\lambda \,x^{2} a -a \lambda \right ) u}{a \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 \lambda x}{x^{2}-1}+\frac {2 \left (\lambda \,x^{2} a -a \lambda \right ) x}{a \left (x^{2}-1\right )^{2}}\\ f_1 f_2 &=\frac {\left (b \,x^{3}-b x \right ) \left (\lambda \,x^{2} a -a \lambda \right )}{a^{2} \left (x^{2}-1\right )^{2}}\\ f_2^2 f_0 &=-\frac {\left (\lambda \,x^{2} a -a \lambda \right )^{2} \left (c \,x^{2}+d x +s \right )}{a^{3} \left (x^{2}-1\right )^{3}} \end {align*}
Substituting the above terms back in equation (2) gives \begin {align*} -\frac {\left (\lambda \,x^{2} a -a \lambda \right ) u^{\prime \prime }\left (x \right )}{a \left (x^{2}-1\right )}-\left (-\frac {2 \lambda x}{x^{2}-1}+\frac {2 \left (\lambda \,x^{2} a -a \lambda \right ) x}{a \left (x^{2}-1\right )^{2}}+\frac {\left (b \,x^{3}-b x \right ) \left (\lambda \,x^{2} a -a \lambda \right )}{a^{2} \left (x^{2}-1\right )^{2}}\right ) u^{\prime }\left (x \right )-\frac {\left (\lambda \,x^{2} a -a \lambda \right )^{2} \left (c \,x^{2}+d x +s \right ) u \left (x \right )}{a^{3} \left (x^{2}-1\right )^{3}} &=0 \end {align*}
Solving the above ODE (this ode solved using Maple, not this program), gives
\[ u \left (x \right ) = \operatorname {DESol}\left (\left \{\textit {\_Y}^{\prime \prime }\left (x \right )-\frac {\left (-x^{3}+x \right ) b \textit {\_Y}^{\prime }\left (x \right )}{a \left (x^{2}-1\right )}-\frac {\left (-c \,x^{2}-d x -s \right ) \lambda \textit {\_Y} \left (x \right )}{a \left (x^{2}-1\right )}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right ) \] The above shows that \[ u^{\prime }\left (x \right ) = \frac {\partial }{\partial x}\operatorname {DESol}\left (\left \{\textit {\_Y}^{\prime \prime }\left (x \right )-\frac {\left (-x^{3}+x \right ) b \textit {\_Y}^{\prime }\left (x \right )}{a \left (x^{2}-1\right )}-\frac {\left (-c \,x^{2}-d x -s \right ) \lambda \textit {\_Y} \left (x \right )}{a \left (x^{2}-1\right )}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right ) \] Using the above in (1) gives the solution \[ y = \frac {\left (\frac {\partial }{\partial x}\operatorname {DESol}\left (\left \{\textit {\_Y}^{\prime \prime }\left (x \right )-\frac {\left (-x^{3}+x \right ) b \textit {\_Y}^{\prime }\left (x \right )}{a \left (x^{2}-1\right )}-\frac {\left (-c \,x^{2}-d x -s \right ) \lambda \textit {\_Y} \left (x \right )}{a \left (x^{2}-1\right )}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )\right ) a \left (x^{2}-1\right )}{\left (\lambda \,x^{2} a -a \lambda \right ) \operatorname {DESol}\left (\left \{\textit {\_Y}^{\prime \prime }\left (x \right )-\frac {\left (-x^{3}+x \right ) b \textit {\_Y}^{\prime }\left (x \right )}{a \left (x^{2}-1\right )}-\frac {\left (-c \,x^{2}-d x -s \right ) \lambda \textit {\_Y} \left (x \right )}{a \left (x^{2}-1\right )}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )} \] Dividing both numerator and denominator by \(c_{1}\) gives, after renaming the constant \(\frac {c_{2}}{c_{1}}=c_{3}\) the following solution
\[ y = \frac {\frac {\partial }{\partial x}\operatorname {DESol}\left (\left \{\frac {\textit {\_Y}^{\prime \prime }\left (x \right ) a \left (x^{2}-1\right )+b \left (x^{3}-x \right ) \textit {\_Y}^{\prime }\left (x \right )+\left (c \,x^{2}+d x +s \right ) \lambda \textit {\_Y} \left (x \right )}{a \left (x^{2}-1\right )}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )}{\lambda \operatorname {DESol}\left (\left \{\frac {\textit {\_Y}^{\prime \prime }\left (x \right ) a \left (x^{2}-1\right )+b \left (x^{3}-x \right ) \textit {\_Y}^{\prime }\left (x \right )+\left (c \,x^{2}+d x +s \right ) \lambda \textit {\_Y} \left (x \right )}{a \left (x^{2}-1\right )}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )} \]
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {\frac {\partial }{\partial x}\operatorname {DESol}\left (\left \{\frac {\textit {\_Y}^{\prime \prime }\left (x \right ) a \left (x^{2}-1\right )+b \left (x^{3}-x \right ) \textit {\_Y}^{\prime }\left (x \right )+\left (c \,x^{2}+d x +s \right ) \lambda \textit {\_Y} \left (x \right )}{a \left (x^{2}-1\right )}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )}{\lambda \operatorname {DESol}\left (\left \{\frac {\textit {\_Y}^{\prime \prime }\left (x \right ) a \left (x^{2}-1\right )+b \left (x^{3}-x \right ) \textit {\_Y}^{\prime }\left (x \right )+\left (c \,x^{2}+d x +s \right ) \lambda \textit {\_Y} \left (x \right )}{a \left (x^{2}-1\right )}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )} \\ \end{align*}
Verification of solutions
\[ y = \frac {\frac {\partial }{\partial x}\operatorname {DESol}\left (\left \{\frac {\textit {\_Y}^{\prime \prime }\left (x \right ) a \left (x^{2}-1\right )+b \left (x^{3}-x \right ) \textit {\_Y}^{\prime }\left (x \right )+\left (c \,x^{2}+d x +s \right ) \lambda \textit {\_Y} \left (x \right )}{a \left (x^{2}-1\right )}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )}{\lambda \operatorname {DESol}\left (\left \{\frac {\textit {\_Y}^{\prime \prime }\left (x \right ) a \left (x^{2}-1\right )+b \left (x^{3}-x \right ) \textit {\_Y}^{\prime }\left (x \right )+\left (c \,x^{2}+d x +s \right ) \lambda \textit {\_Y} \left (x \right )}{a \left (x^{2}-1\right )}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & a \left (x^{2}-1\right ) \left (y^{\prime }+\lambda y^{2}\right )+b x \left (x^{2}-1\right ) y=-c \,x^{2}-d x -s \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-\frac {y^{2} a \lambda \,x^{2}+b \,x^{3} y-y^{2} a \lambda -y b x +c \,x^{2}+d x +s}{a \left (x^{2}-1\right )} \end {array} \]
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) = -b*x*(diff(y(x), x))/a-lambda*(c*x^2+d*x+s)*y(x)/(a*(x^2-1)), y(x)` 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 -> hyper3: Equivalence to 2F1, 1F1 or 0F1 under a power @ Moebius -> Mathieu -> Equivalence to the rational form of Mathieu ODE under a power @ Moebius trying a solution in terms of MeijerG functions -> Heun: Equivalence to the GHE or one of its 4 confluent cases under a power @ Moebius -> trying a solution of the form r0(x) * Y + r1(x) * Y where Y = exp(int(r(x), dx)) * 2F1([a1, a2], [b1], f) trying a symmetry of the form [xi=0, eta=F(x)] trying 2nd order exact linear trying symmetries linear in x and y(x) trying to convert to a linear ODE with constant coefficients trying 2nd order, integrating factor of the form mu(x,y) -> Trying a solution in terms of special functions: -> Bessel -> elliptic -> Legendre -> Whittaker -> hyper3: Equivalence to 1F1 under a power @ Moebius -> hypergeometric -> heuristic approach -> hyper3: Equivalence to 2F1, 1F1 or 0F1 under a power @ Moebius -> Mathieu -> Equivalence to the rational form of Mathieu ODE under a power @ Moebius trying 2nd order exact linear trying symmetries linear in x and y(x) trying to convert to a linear ODE with constant coefficients trying to convert to an ODE of Bessel type trying to convert to an ODE of Bessel type -> Trying a change of variables to reduce to Bernoulli -> Calling odsolve with the ODE`, diff(y(x), x)-((-lambda*x^2/(x^2-1)+lambda/(x^2-1))*y(x)^2+y(x)+(-b*x^3/(a*(x^2-1))+b*x/(a*(x^2 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 inverse_Riccati trying 1st order ODE linearizable_by_differentiation -> trying a symmetry pattern of the form [F(x)*G(y), 0] -> trying a symmetry pattern of the form [0, F(x)*G(y)] -> trying a symmetry pattern of the form [F(x),G(x)*y+H(x)] trying inverse_Riccati trying 1st order ODE linearizable_by_differentiation --- Trying Lie symmetry methods, 1st order --- `, `-> Computing symmetries using: way = 4 `, `-> Computing symmetries using: way = 2 `, `-> Computing symmetries using: way = 6`
✗ Solution by Maple
dsolve(a*(x^2-1)*(diff(y(x),x)+lambda*y(x)^2)+b*x*(x^2-1)*y(x)+c*x^2+d*x+s=0,y(x), singsol=all)
\[ \text {No solution found} \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[a*(x^2-1)*(y'[x]+\[Lambda]*y[x]^2)+b*x*(x^2-1)*y[x]+c*x^2+d*x+s==0,y[x],x,IncludeSingularSolutions -> True]
Not solved