2.77 problem 77

Internal problem ID [10406]
Internal file name [OUTPUT/9354_Monday_June_06_2022_02_16_53_PM_78079711/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: 77.
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]

Unable to solve or complete the solution.

\[ \boxed {\left (x^{n} a +b \,x^{m}+c \right ) y^{\prime }-\alpha \,x^{k} y^{2}-\beta \,x^{s} y=-\alpha \,\lambda ^{2} x^{k}+\beta \lambda \,x^{s}} \]

2.77.1 Solving as riccati ode

In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= \frac {\alpha \,x^{k} y^{2}+\beta \,x^{s} y -\alpha \,\lambda ^{2} x^{k}+\beta \lambda \,x^{s}}{x^{n} a +b \,x^{m}+c} \end {align*}

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

Substituting the above terms back in equation (2) gives \begin {align*} \frac {\alpha \,x^{k} u^{\prime \prime }\left (x \right )}{x^{n} a +b \,x^{m}+c}-\left (\frac {\alpha k \,x^{k}}{x \left (x^{n} a +b \,x^{m}+c \right )}-\frac {\alpha \,x^{k} \left (\frac {x^{n} n a}{x}+\frac {b \,x^{m} m}{x}\right )}{\left (x^{n} a +b \,x^{m}+c \right )^{2}}+\frac {x^{s} \beta \alpha \,x^{k}}{\left (x^{n} a +b \,x^{m}+c \right )^{2}}\right ) u^{\prime }\left (x \right )+\frac {\alpha ^{2} x^{2 k} \left (-\alpha \,\lambda ^{2} x^{k}+\beta \lambda \,x^{s}\right ) u \left (x \right )}{\left (x^{n} a +b \,x^{m}+c \right )^{3}} &=0 \end {align*}

Solving the above ODE (this ode solved using Maple, not this program), gives Unable to solve. Terminating.

2.77.2 Maple step by step solution

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

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_symmetries 
   trying Riccati to 2nd Order 
   -> Calling odsolve with the ODE`, diff(diff(y(x), x), x) = (beta*x^s*x+x^n*a*k-a*n*x^n+x^m*b*k-b*m*x^m+c*k)*(diff(y(x), x))/(x*(x 
      Methods for second order ODEs: 
      --- Trying classification methods --- 
      trying a symmetry of the form [xi=0, eta=F(x)] 
      checking if the LODE is missing y 
      -> Trying an equivalence, under non-integer power transformations, 
         to LODEs admitting Liouvillian solutions. 
      -> 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 
      -> Heun: Equivalence to the GHE or one of its 4 confluent cases under a power @ Moebius 
      -> 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 changes of variables to rationalize or make the ODE simpler 
      <- unable to find a useful change of variables 
         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 symmetry of the form [xi=0, eta=F(x)] 
         checking if the LODE is missing y 
         -> Trying an equivalence, under non-integer power transformations, 
            to LODEs admitting Liouvillian solutions. 
         -> 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 
         -> Heun: Equivalence to the GHE or one of its 4 confluent cases under a power @ Moebius 
         -> 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 changes of variables to rationalize or make the ODE simpler 
         <- unable to find a useful change of variables 
            trying a symmetry of the form [xi=0, eta=F(x)] 
         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)-(x^k*alpha*y(x)^2/(x^n*a+b*x^m+c)+y(x)+beta*x^s*y(x)*x/(x^n*a+b*x^m+c)+x^2*(-alph 
      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_symmetries 
      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)] 
   <- symmetry pattern of the form [0, F(x)*G(y)] successful 
   <- Riccati with symmetry pattern of the form [0,F(x)*G(y)] successful`
 

✓ Solution by Maple

Time used: 0.015 (sec). Leaf size: 164

dsolve((a*x^n+b*x^m+c)*diff(y(x),x)=alpha*x^k*y(x)^2+beta*x^s*y(x)-alpha*lambda^2*x^k+beta*lambda*x^s,y(x), singsol=all)
 

\[ y \left (x \right ) = \frac {-\alpha \left (\int \frac {x^{k} {\mathrm e}^{-\left (\int \frac {2 \alpha \,x^{k} \lambda -x^{s} \beta }{a \,x^{n}+b \,x^{m}+c}d x \right )}}{a \,x^{n}+b \,x^{m}+c}d x \right ) \lambda -\lambda c_{1} -{\mathrm e}^{-\left (\int \frac {2 \alpha \,x^{k} \lambda -x^{s} \beta }{a \,x^{n}+b \,x^{m}+c}d x \right )}}{c_{1} +\alpha \left (\int \frac {x^{k} {\mathrm e}^{-\left (\int \frac {2 \alpha \,x^{k} \lambda -x^{s} \beta }{a \,x^{n}+b \,x^{m}+c}d x \right )}}{a \,x^{n}+b \,x^{m}+c}d x \right )} \]

✓ Solution by Mathematica

Time used: 13.649 (sec). Leaf size: 389

DSolve[(a*x^n+b*x^m+c)*y'[x]==\[Alpha]*x^k*y[x]^2+\[Beta]*x^s*y[x]-\[Alpha]*\[Lambda]^2*x^k+\[Beta]*\[Lambda]*x^s,y[x],x,IncludeSingularSolutions -> True]
 

\[ \text {Solve}\left [\int _1^x\frac {\exp \left (-\int _1^{K[2]}-\frac {\beta K[1]^s-2 \alpha \lambda K[1]^k}{b K[1]^m+a K[1]^n+c}dK[1]\right ) \left (-\alpha \lambda K[2]^k+\alpha y(x) K[2]^k+\beta K[2]^s\right )}{(k-s) \alpha \beta \left (b K[2]^m+a K[2]^n+c\right ) (\lambda +y(x))}dK[2]+\int _1^{y(x)}\left (-\int _1^x\left (\frac {\exp \left (-\int _1^{K[2]}-\frac {\beta K[1]^s-2 \alpha \lambda K[1]^k}{b K[1]^m+a K[1]^n+c}dK[1]\right ) K[2]^k}{(k-s) \beta \left (b K[2]^m+a K[2]^n+c\right ) (\lambda +K[3])}-\frac {\exp \left (-\int _1^{K[2]}-\frac {\beta K[1]^s-2 \alpha \lambda K[1]^k}{b K[1]^m+a K[1]^n+c}dK[1]\right ) \left (-\alpha \lambda K[2]^k+\alpha K[3] K[2]^k+\beta K[2]^s\right )}{(k-s) \alpha \beta \left (b K[2]^m+a K[2]^n+c\right ) (\lambda +K[3])^2}\right )dK[2]-\frac {\exp \left (-\int _1^x-\frac {\beta K[1]^s-2 \alpha \lambda K[1]^k}{b K[1]^m+a K[1]^n+c}dK[1]\right )}{(k-s) \alpha \beta (\lambda +K[3])^2}\right )dK[3]=c_1,y(x)\right ] \]