Internal problem ID [10360]
Internal file name [OUTPUT/9308_Monday_June_06_2022_01_51_00_PM_6355186/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: 31.
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]
\[ \boxed {y^{\prime }-a \,x^{n} y^{2}+a \,x^{n} \left (b \,x^{m}+c \right ) y=b m \,x^{m -1}} \]
In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= -x^{n} x^{m} a b y -x^{n} a c y +a \,x^{n} y^{2}+b m \,x^{m -1} \end {align*}
This is a Riccati ODE. Comparing the ODE to solve \[ y' = -x^{n} x^{m} a b y -x^{n} a c y +a \,x^{n} y^{2}+\frac {b \,x^{m} m}{x} \] With Riccati ODE standard form \[ y' = f_0(x)+ f_1(x)y+f_2(x)y^{2} \] Shows that \(f_0(x)=b m \,x^{m -1}\), \(f_1(x)=-b \,x^{m} x^{n} a -x^{n} a c\) 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 &=\left (-b \,x^{m} x^{n} a -x^{n} a c \right ) x^{n} a\\ f_2^2 f_0 &=x^{2 n} a^{2} b m \,x^{m -1} \end {align*}
Substituting the above terms back in equation (2) gives \begin {align*} x^{n} a u^{\prime \prime }\left (x \right )-\left (\left (-b \,x^{m} x^{n} a -x^{n} a c \right ) x^{n} a +\frac {x^{n} n a}{x}\right ) u^{\prime }\left (x \right )+x^{2 n} a^{2} b m \,x^{m -1} u \left (x \right ) &=0 \end {align*}
Solving the above ODE (this ode solved using Maple, not this program), gives
\[ u \left (x \right ) = \operatorname {DESol}\left (\left \{\frac {a b m \,x^{m +n} \textit {\_Y} \left (x \right )+x^{1+m +n} \textit {\_Y}^{\prime }\left (x \right ) a b +x^{1+n} \textit {\_Y}^{\prime }\left (x \right ) a c +\textit {\_Y}^{\prime \prime }\left (x \right ) x -n \textit {\_Y}^{\prime }\left (x \right )}{x}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right ) \] The above shows that \[ u^{\prime }\left (x \right ) = \frac {\partial }{\partial x}\operatorname {DESol}\left (\left \{\frac {a b m \,x^{m +n} \textit {\_Y} \left (x \right )+x^{1+m +n} \textit {\_Y}^{\prime }\left (x \right ) a b +x^{1+n} \textit {\_Y}^{\prime }\left (x \right ) a c +\textit {\_Y}^{\prime \prime }\left (x \right ) x -n \textit {\_Y}^{\prime }\left (x \right )}{x}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right ) \] Using the above in (1) gives the solution \[ y = -\frac {\left (\frac {\partial }{\partial x}\operatorname {DESol}\left (\left \{\frac {a b m \,x^{m +n} \textit {\_Y} \left (x \right )+x^{1+m +n} \textit {\_Y}^{\prime }\left (x \right ) a b +x^{1+n} \textit {\_Y}^{\prime }\left (x \right ) a c +\textit {\_Y}^{\prime \prime }\left (x \right ) x -n \textit {\_Y}^{\prime }\left (x \right )}{x}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )\right ) x^{-n}}{a \operatorname {DESol}\left (\left \{\frac {a b m \,x^{m +n} \textit {\_Y} \left (x \right )+x^{1+m +n} \textit {\_Y}^{\prime }\left (x \right ) a b +x^{1+n} \textit {\_Y}^{\prime }\left (x \right ) a c +\textit {\_Y}^{\prime \prime }\left (x \right ) x -n \textit {\_Y}^{\prime }\left (x \right )}{x}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )} \] Dividing both numerator and denominator by \(c_{1}\) gives, after renaming the constant \(\frac {c_{2}}{c_{1}}=c_{3}\) the following solution
\[ y = -\frac {\left (\frac {\partial }{\partial x}\operatorname {DESol}\left (\left \{\frac {a b m \,x^{m +n} \textit {\_Y} \left (x \right )+x^{1+m +n} \textit {\_Y}^{\prime }\left (x \right ) a b +x^{1+n} \textit {\_Y}^{\prime }\left (x \right ) a c +\textit {\_Y}^{\prime \prime }\left (x \right ) x -n \textit {\_Y}^{\prime }\left (x \right )}{x}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )\right ) x^{-n}}{a \operatorname {DESol}\left (\left \{\frac {a b m \,x^{m +n} \textit {\_Y} \left (x \right )+x^{1+m +n} \textit {\_Y}^{\prime }\left (x \right ) a b +x^{1+n} \textit {\_Y}^{\prime }\left (x \right ) a c +\textit {\_Y}^{\prime \prime }\left (x \right ) x -n \textit {\_Y}^{\prime }\left (x \right )}{x}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )} \]
The solution(s) found are the following \begin{align*} \tag{1} y &= -\frac {\left (\frac {\partial }{\partial x}\operatorname {DESol}\left (\left \{\frac {a b m \,x^{m +n} \textit {\_Y} \left (x \right )+x^{1+m +n} \textit {\_Y}^{\prime }\left (x \right ) a b +x^{1+n} \textit {\_Y}^{\prime }\left (x \right ) a c +\textit {\_Y}^{\prime \prime }\left (x \right ) x -n \textit {\_Y}^{\prime }\left (x \right )}{x}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )\right ) x^{-n}}{a \operatorname {DESol}\left (\left \{\frac {a b m \,x^{m +n} \textit {\_Y} \left (x \right )+x^{1+m +n} \textit {\_Y}^{\prime }\left (x \right ) a b +x^{1+n} \textit {\_Y}^{\prime }\left (x \right ) a c +\textit {\_Y}^{\prime \prime }\left (x \right ) x -n \textit {\_Y}^{\prime }\left (x \right )}{x}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )} \\ \end{align*}
Verification of solutions
\[ y = -\frac {\left (\frac {\partial }{\partial x}\operatorname {DESol}\left (\left \{\frac {a b m \,x^{m +n} \textit {\_Y} \left (x \right )+x^{1+m +n} \textit {\_Y}^{\prime }\left (x \right ) a b +x^{1+n} \textit {\_Y}^{\prime }\left (x \right ) a c +\textit {\_Y}^{\prime \prime }\left (x \right ) x -n \textit {\_Y}^{\prime }\left (x \right )}{x}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )\right ) x^{-n}}{a \operatorname {DESol}\left (\left \{\frac {a b m \,x^{m +n} \textit {\_Y} \left (x \right )+x^{1+m +n} \textit {\_Y}^{\prime }\left (x \right ) a b +x^{1+n} \textit {\_Y}^{\prime }\left (x \right ) a c +\textit {\_Y}^{\prime \prime }\left (x \right ) x -n \textit {\_Y}^{\prime }\left (x \right )}{x}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-a \,x^{n} y^{2}+a \,x^{n} \left (b \,x^{m}+c \right ) y=b m \,x^{m -1} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=a \,x^{n} y^{2}-a \,x^{n} \left (b \,x^{m}+c \right ) y+b m \,x^{m -1} \end {array} \]
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) = -(a*x^n*c*x+x^(n+m)*a*b*x-n)*(diff(y(x), x))/x-a*x^n*b*m*x^(m-1)*y(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 -> 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)+(-x^(n+m)*a*b-a*x^n*c)*y(x)*x+x^2*b*m*x^(m-1))/x, y(x), explic 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-a*x^n*(b*x^m+c)*y(x)+b*m*x^(m-1),y(x), singsol=all)
\[ \text {No solution found} \]
✓ Solution by Mathematica
Time used: 59.342 (sec). Leaf size: 353
DSolve[y'[x]==a*x^n*y[x]^2-a*x^n*(b*x^m+c)*y[x]+b*m*x^(m-1),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {b m \left (b x^m+c\right )^2 \left (1+c_1 \int _1^x\frac {\exp \left (a K[1]^{n+1} \left (\frac {b K[1]^m}{m+n+1}+\frac {c}{n+1}\right )\right ) K[1]^{m-1}}{\left (b K[1]^m+c\right )^2}dK[1]\right )}{b c_1 m \left (b x^m+c\right ) \int _1^x\frac {\exp \left (a K[1]^{n+1} \left (\frac {b K[1]^m}{m+n+1}+\frac {c}{n+1}\right )\right ) K[1]^{m-1}}{\left (b K[1]^m+c\right )^2}dK[1]+c_1 e^{a x^{n+1} \left (\frac {b x^m}{m+n+1}+\frac {c}{n+1}\right )}+b^2 m x^m+b c m} \\ y(x)\to \frac {b m \left (b x^m+c\right )^2 \int _1^x\frac {\exp \left (a K[1]^{n+1} \left (\frac {b K[1]^m}{m+n+1}+\frac {c}{n+1}\right )\right ) K[1]^{m-1}}{\left (b K[1]^m+c\right )^2}dK[1]}{b m \left (b x^m+c\right ) \int _1^x\frac {\exp \left (a K[1]^{n+1} \left (\frac {b K[1]^m}{m+n+1}+\frac {c}{n+1}\right )\right ) K[1]^{m-1}}{\left (b K[1]^m+c\right )^2}dK[1]+e^{a x^{n+1} \left (\frac {b x^m}{m+n+1}+\frac {c}{n+1}\right )}} \\ \end{align*}