2.33 problem 33

Internal problem ID [10362]
Internal file name [OUTPUT/9310_Monday_June_06_2022_01_51_07_PM_78619454/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: 33.
ODE order: 1.
ODE degree: 1.

The type(s) of ODE detected by this program : "riccati"

Maple gives the following as the ode type

[_Riccati]

\[ \boxed {y^{\prime }-a \,x^{n} y^{2}-b \,x^{m} y=c k \,x^{k -1}-b c \,x^{k +m}-a \,c^{2} x^{n +2 k}} \]

2.33.1 Solving as riccati ode

In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= a \,x^{n} y^{2}+x^{m} b y +c k \,x^{k -1}-b c \,x^{k +m}-a \,c^{2} x^{n +2 k} \end {align*}

This is a Riccati ODE. Comparing the ODE to solve \[ y' = a \,x^{n} y^{2}+x^{m} b y +\frac {c k \,x^{k}}{x}-b c \,x^{k} x^{m}-a \,c^{2} x^{n} x^{2 k} \] With Riccati ODE standard form \[ y' = f_0(x)+ f_1(x)y+f_2(x)y^{2} \] Shows that \(f_0(x)=c k \,x^{k -1}-b c \,x^{k +m}-a \,c^{2} x^{n +2 k}\), \(f_1(x)=b \,x^{m}\) and \(f_2(x)=x^{n} a\). Let \begin {align*} y &= \frac {-u'}{f_2 u} \\ &= \frac {-u'}{x^{n} a u} \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 {x^{n} n a}{x}\\ f_1 f_2 &=b \,x^{m} x^{n} a\\ f_2^2 f_0 &=x^{2 n} a^{2} \left (c k \,x^{k -1}-b c \,x^{k +m}-a \,c^{2} x^{n +2 k}\right ) \end {align*}

Substituting the above terms back in equation (2) gives \begin {align*} x^{n} a u^{\prime \prime }\left (x \right )-\left (\frac {x^{n} n a}{x}+b \,x^{m} x^{n} a \right ) u^{\prime }\left (x \right )+x^{2 n} a^{2} \left (c k \,x^{k -1}-b c \,x^{k +m}-a \,c^{2} x^{n +2 k}\right ) u \left (x \right ) &=0 \end {align*}

Solving the above ODE (this ode solved using Maple, not this program), gives

\[ u \left (x \right ) = \operatorname {DESol}\left (\left \{\textit {\_Y}^{\prime \prime }\left (x \right )-\textit {\_Y}^{\prime }\left (x \right ) \left (\frac {n}{x}+b \,x^{m}\right )+x^{n} a \left (c k \,x^{k -1}-b c \,x^{k +m}-a \,c^{2} x^{n +2 k}\right ) \textit {\_Y} \left (x \right )\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right ) \] The above shows that \[ u^{\prime }\left (x \right ) = \frac {\partial }{\partial x}\operatorname {DESol}\left (\left \{\textit {\_Y}^{\prime \prime }\left (x \right )-\textit {\_Y}^{\prime }\left (x \right ) \left (\frac {n}{x}+b \,x^{m}\right )+x^{n} a \left (c k \,x^{k -1}-b c \,x^{k +m}-a \,c^{2} x^{n +2 k}\right ) \textit {\_Y} \left (x \right )\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right ) \] Using the above in (1) gives the solution \[ y = -\frac {\left (\frac {\partial }{\partial x}\operatorname {DESol}\left (\left \{\textit {\_Y}^{\prime \prime }\left (x \right )-\textit {\_Y}^{\prime }\left (x \right ) \left (\frac {n}{x}+b \,x^{m}\right )+x^{n} a \left (c k \,x^{k -1}-b c \,x^{k +m}-a \,c^{2} x^{n +2 k}\right ) \textit {\_Y} \left (x \right )\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )\right ) x^{-n}}{a \operatorname {DESol}\left (\left \{\textit {\_Y}^{\prime \prime }\left (x \right )-\textit {\_Y}^{\prime }\left (x \right ) \left (\frac {n}{x}+b \,x^{m}\right )+x^{n} a \left (c k \,x^{k -1}-b c \,x^{k +m}-a \,c^{2} x^{n +2 k}\right ) \textit {\_Y} \left (x \right )\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )} \] Dividing both numerator and denominator by \(c_{1}\) gives, after renaming the constant \(\frac {c_{2}}{c_{1}}=c_{3}\) the following solution

\[ y = -\frac {\left (\frac {\partial }{\partial x}\operatorname {DESol}\left (\left \{\frac {-x^{n +2 k} x^{1+n} \textit {\_Y} \left (x \right ) a^{2} c^{2}+a c \textit {\_Y} \left (x \right ) \left (k \,x^{k -1}-x^{k +m} b \right ) x^{1+n}+\textit {\_Y}^{\prime \prime }\left (x \right ) x -\textit {\_Y}^{\prime }\left (x \right ) \left (b \,x^{m +1}+n \right )}{x}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )\right ) x^{-n}}{a \operatorname {DESol}\left (\left \{\frac {-x^{2 n +2 k +1} \textit {\_Y} \left (x \right ) a^{2} c^{2}-x^{k +1+m +n} \textit {\_Y} \left (x \right ) a b c +\textit {\_Y}^{\prime \prime }\left (x \right ) x +x^{k +n} \textit {\_Y} \left (x \right ) a c k -\textit {\_Y}^{\prime }\left (x \right ) \left (b \,x^{m +1}+n \right )}{x}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )} \]

Summary

The solution(s) found are the following \begin{align*} \tag{1} y &= -\frac {\left (\frac {\partial }{\partial x}\operatorname {DESol}\left (\left \{\frac {-x^{n +2 k} x^{1+n} \textit {\_Y} \left (x \right ) a^{2} c^{2}+a c \textit {\_Y} \left (x \right ) \left (k \,x^{k -1}-x^{k +m} b \right ) x^{1+n}+\textit {\_Y}^{\prime \prime }\left (x \right ) x -\textit {\_Y}^{\prime }\left (x \right ) \left (b \,x^{m +1}+n \right )}{x}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )\right ) x^{-n}}{a \operatorname {DESol}\left (\left \{\frac {-x^{2 n +2 k +1} \textit {\_Y} \left (x \right ) a^{2} c^{2}-x^{k +1+m +n} \textit {\_Y} \left (x \right ) a b c +\textit {\_Y}^{\prime \prime }\left (x \right ) x +x^{k +n} \textit {\_Y} \left (x \right ) a c k -\textit {\_Y}^{\prime }\left (x \right ) \left (b \,x^{m +1}+n \right )}{x}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )} \\ \end{align*}

Verification of solutions

\[ y = -\frac {\left (\frac {\partial }{\partial x}\operatorname {DESol}\left (\left \{\frac {-x^{n +2 k} x^{1+n} \textit {\_Y} \left (x \right ) a^{2} c^{2}+a c \textit {\_Y} \left (x \right ) \left (k \,x^{k -1}-x^{k +m} b \right ) x^{1+n}+\textit {\_Y}^{\prime \prime }\left (x \right ) x -\textit {\_Y}^{\prime }\left (x \right ) \left (b \,x^{m +1}+n \right )}{x}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )\right ) x^{-n}}{a \operatorname {DESol}\left (\left \{\frac {-x^{2 n +2 k +1} \textit {\_Y} \left (x \right ) a^{2} c^{2}-x^{k +1+m +n} \textit {\_Y} \left (x \right ) a b c +\textit {\_Y}^{\prime \prime }\left (x \right ) x +x^{k +n} \textit {\_Y} \left (x \right ) a c k -\textit {\_Y}^{\prime }\left (x \right ) \left (b \,x^{m +1}+n \right )}{x}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )} \] Verified OK.

2.33.2 Maple step by step solution

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

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) = (x*x^m*b+n)*(diff(y(x), x))/x-a*x^n*c*(-a*c*x^(n+2*k)+x^(-1+k)*k-x^(m+ 
      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 
         -> Kummer 
            -> 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)-(a*x^n*y(x)^2+y(x)+b*x^m*y(x)*x+x^2*(c*k*x^(-1+k)-b*c*x^(m+k)-a*c^2*x^(n+2*k)))/x 
      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)] 
   -> 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(diff(y(x),x)=a*x^n*y(x)^2+b*x^m*y(x)+c*k*x^(k-1)-b*c*x^(m+k)-a*c^2*x^(n+2*k),y(x), singsol=all)
 

\[ \text {No solution found} \]

✗ Solution by Mathematica

Time used: 0.0 (sec). Leaf size: 0

DSolve[y'[x]==a*x^n*y[x]^2+b*x^m*y[x]+c*k*x^(k-1)-b*c*x^(m+k)-a*c^2*x^(n+2*k),y[x],x,IncludeSingularSolutions -> True]
 

Not solved