Internal problem ID [10405]
Internal file name [OUTPUT/9353_Monday_June_06_2022_02_16_17_PM_53182186/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: 76.
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]
\[ \boxed {\left (x^{n} a +b \,x^{m}+c \right ) y^{\prime }-a \,x^{-2+n} y^{2}-b \,x^{m -1} y=c} \]
In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= \frac {a \,x^{-2+n} y^{2}+b \,x^{m -1} y +c}{x^{n} a +b \,x^{m}+c} \end {align*}
This is a Riccati ODE. Comparing the ODE to solve \[ y' = \frac {a \,x^{n} y^{2}}{\left (x^{n} a +b \,x^{m}+c \right ) x^{2}}+\frac {b \,x^{m} y}{\left (x^{n} a +b \,x^{m}+c \right ) x}+\frac {c}{x^{n} a +b \,x^{m}+c} \] With Riccati ODE standard form \[ y' = f_0(x)+ f_1(x)y+f_2(x)y^{2} \] Shows that \(f_0(x)=\frac {c}{x^{n} a +b \,x^{m}+c}\), \(f_1(x)=\frac {b \,x^{m -1}}{x^{n} a +b \,x^{m}+c}\) and \(f_2(x)=\frac {a \,x^{-2+n}}{x^{n} a +b \,x^{m}+c}\). Let \begin {align*} y &= \frac {-u'}{f_2 u} \\ &= \frac {-u'}{\frac {a \,x^{-2+n} 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 {a \,x^{-2+n} \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 {a \,x^{-2+n} \left (-2+n \right )}{\left (x^{n} a +b \,x^{m}+c \right ) x}\\ f_1 f_2 &=\frac {b \,x^{m -1} a \,x^{-2+n}}{\left (x^{n} a +b \,x^{m}+c \right )^{2}}\\ f_2^2 f_0 &=\frac {a^{2} x^{-4+2 n} c}{\left (x^{n} a +b \,x^{m}+c \right )^{3}} \end {align*}
Substituting the above terms back in equation (2) gives \begin {align*} \frac {a \,x^{-2+n} u^{\prime \prime }\left (x \right )}{x^{n} a +b \,x^{m}+c}-\left (-\frac {a \,x^{-2+n} \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 {a \,x^{-2+n} \left (-2+n \right )}{\left (x^{n} a +b \,x^{m}+c \right ) x}+\frac {b \,x^{m -1} a \,x^{-2+n}}{\left (x^{n} a +b \,x^{m}+c \right )^{2}}\right ) u^{\prime }\left (x \right )+\frac {a^{2} x^{-4+2 n} c 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
\[ u \left (x \right ) = \operatorname {DESol}\left (\left \{\frac {\textit {\_Y}^{\prime \prime }\left (x \right ) \left (x^{n} a +b \,x^{m}+c \right )^{2} x +a c \,x^{-2+n} \textit {\_Y} \left (x \right ) x +2 \left (\frac {b \left (m -n +1\right ) x^{m}}{2}+x^{n} a -\frac {c \left (-2+n \right )}{2}\right ) \textit {\_Y}^{\prime }\left (x \right ) \left (x^{n} a +b \,x^{m}+c \right )}{x \left (x^{n} a +b \,x^{m}+c \right )^{2}}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right ) \] The above shows that \[ u^{\prime }\left (x \right ) = \frac {\partial }{\partial x}\operatorname {DESol}\left (\left \{\frac {\textit {\_Y}^{\prime \prime }\left (x \right ) \left (x^{n} a +b \,x^{m}+c \right )^{2} x +a c \,x^{-1+n} \textit {\_Y} \left (x \right )+2 \left (\frac {b \left (m -n +1\right ) x^{m}}{2}+x^{n} a -\frac {c \left (-2+n \right )}{2}\right ) \textit {\_Y}^{\prime }\left (x \right ) \left (x^{n} a +b \,x^{m}+c \right )}{x \left (x^{n} a +b \,x^{m}+c \right )^{2}}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right ) \] Using the above in (1) gives the solution \[ y = -\frac {\left (\frac {\partial }{\partial x}\operatorname {DESol}\left (\left \{\frac {\textit {\_Y}^{\prime \prime }\left (x \right ) \left (x^{n} a +b \,x^{m}+c \right )^{2} x +a c \,x^{-1+n} \textit {\_Y} \left (x \right )+2 \left (\frac {b \left (m -n +1\right ) x^{m}}{2}+x^{n} a -\frac {c \left (-2+n \right )}{2}\right ) \textit {\_Y}^{\prime }\left (x \right ) \left (x^{n} a +b \,x^{m}+c \right )}{x \left (x^{n} a +b \,x^{m}+c \right )^{2}}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )\right ) \left (x^{n} a +b \,x^{m}+c \right ) x^{2-n}}{a \operatorname {DESol}\left (\left \{\frac {\textit {\_Y}^{\prime \prime }\left (x \right ) \left (x^{n} a +b \,x^{m}+c \right )^{2} x +a c \,x^{-2+n} \textit {\_Y} \left (x \right ) x +2 \left (\frac {b \left (m -n +1\right ) x^{m}}{2}+x^{n} a -\frac {c \left (-2+n \right )}{2}\right ) \textit {\_Y}^{\prime }\left (x \right ) \left (x^{n} a +b \,x^{m}+c \right )}{x \left (x^{n} a +b \,x^{m}+c \right )^{2}}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )} \] Dividing both numerator and denominator by \(c_{1}\) gives, after renaming the constant \(\frac {c_{2}}{c_{1}}=c_{3}\) the following solution
\[ y = -\frac {x^{2-n} \left (x^{n} a +b \,x^{m}+c \right ) \left (\frac {\partial }{\partial x}\operatorname {DESol}\left (\left \{\frac {x^{1+2 m} b^{2} \textit {\_Y}^{\prime \prime }\left (x \right )+x^{1+2 n} a^{2} \textit {\_Y}^{\prime \prime }\left (x \right )+2 a b \,x^{1+m +n} \textit {\_Y}^{\prime \prime }\left (x \right )+2 \left (a \,x^{1+n}+b \,x^{m +1}+\frac {c x}{2}\right ) c \textit {\_Y}^{\prime \prime }\left (x \right )+a c \,x^{-1+n} \textit {\_Y} \left (x \right )+2 \textit {\_Y}^{\prime }\left (x \right ) \left (\frac {a b \,x^{m +n} \left (m -n +3\right )}{2}+\frac {b^{2} \left (m -n +1\right ) x^{2 m}}{2}+a^{2} x^{2 n}-\frac {\left (-b \left (m -2 n +3\right ) x^{m}+\left (n -4\right ) a \,x^{n}+c \left (-2+n \right )\right ) c}{2}\right )}{x \left (x^{n} a +b \,x^{m}+c \right )^{2}}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )\right )}{a \operatorname {DESol}\left (\left \{\frac {x^{2} \left (x^{n} a +b \,x^{m}+c \right )^{2} \textit {\_Y}^{\prime \prime }\left (x \right )+2 \left (\frac {b \left (m -n +1\right ) x^{m}}{2}+x^{n} a -\frac {c \left (-2+n \right )}{2}\right ) \left (x^{n} a +b \,x^{m}+c \right ) x \textit {\_Y}^{\prime }\left (x \right )+a c \,x^{n} \textit {\_Y} \left (x \right )}{x^{2} \left (x^{n} a +b \,x^{m}+c \right )^{2}}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )} \]
The solution(s) found are the following \begin{align*} \tag{1} y &= -\frac {x^{2-n} \left (x^{n} a +b \,x^{m}+c \right ) \left (\frac {\partial }{\partial x}\operatorname {DESol}\left (\left \{\frac {x^{1+2 m} b^{2} \textit {\_Y}^{\prime \prime }\left (x \right )+x^{1+2 n} a^{2} \textit {\_Y}^{\prime \prime }\left (x \right )+2 a b \,x^{1+m +n} \textit {\_Y}^{\prime \prime }\left (x \right )+2 \left (a \,x^{1+n}+b \,x^{m +1}+\frac {c x}{2}\right ) c \textit {\_Y}^{\prime \prime }\left (x \right )+a c \,x^{-1+n} \textit {\_Y} \left (x \right )+2 \textit {\_Y}^{\prime }\left (x \right ) \left (\frac {a b \,x^{m +n} \left (m -n +3\right )}{2}+\frac {b^{2} \left (m -n +1\right ) x^{2 m}}{2}+a^{2} x^{2 n}-\frac {\left (-b \left (m -2 n +3\right ) x^{m}+\left (n -4\right ) a \,x^{n}+c \left (-2+n \right )\right ) c}{2}\right )}{x \left (x^{n} a +b \,x^{m}+c \right )^{2}}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )\right )}{a \operatorname {DESol}\left (\left \{\frac {x^{2} \left (x^{n} a +b \,x^{m}+c \right )^{2} \textit {\_Y}^{\prime \prime }\left (x \right )+2 \left (\frac {b \left (m -n +1\right ) x^{m}}{2}+x^{n} a -\frac {c \left (-2+n \right )}{2}\right ) \left (x^{n} a +b \,x^{m}+c \right ) x \textit {\_Y}^{\prime }\left (x \right )+a c \,x^{n} \textit {\_Y} \left (x \right )}{x^{2} \left (x^{n} a +b \,x^{m}+c \right )^{2}}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )} \\ \end{align*}
Verification of solutions
\[ y = -\frac {x^{2-n} \left (x^{n} a +b \,x^{m}+c \right ) \left (\frac {\partial }{\partial x}\operatorname {DESol}\left (\left \{\frac {x^{1+2 m} b^{2} \textit {\_Y}^{\prime \prime }\left (x \right )+x^{1+2 n} a^{2} \textit {\_Y}^{\prime \prime }\left (x \right )+2 a b \,x^{1+m +n} \textit {\_Y}^{\prime \prime }\left (x \right )+2 \left (a \,x^{1+n}+b \,x^{m +1}+\frac {c x}{2}\right ) c \textit {\_Y}^{\prime \prime }\left (x \right )+a c \,x^{-1+n} \textit {\_Y} \left (x \right )+2 \textit {\_Y}^{\prime }\left (x \right ) \left (\frac {a b \,x^{m +n} \left (m -n +3\right )}{2}+\frac {b^{2} \left (m -n +1\right ) x^{2 m}}{2}+a^{2} x^{2 n}-\frac {\left (-b \left (m -2 n +3\right ) x^{m}+\left (n -4\right ) a \,x^{n}+c \left (-2+n \right )\right ) c}{2}\right )}{x \left (x^{n} a +b \,x^{m}+c \right )^{2}}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )\right )}{a \operatorname {DESol}\left (\left \{\frac {x^{2} \left (x^{n} a +b \,x^{m}+c \right )^{2} \textit {\_Y}^{\prime \prime }\left (x \right )+2 \left (\frac {b \left (m -n +1\right ) x^{m}}{2}+x^{n} a -\frac {c \left (-2+n \right )}{2}\right ) \left (x^{n} a +b \,x^{m}+c \right ) x \textit {\_Y}^{\prime }\left (x \right )+a c \,x^{n} \textit {\_Y} \left (x \right )}{x^{2} \left (x^{n} a +b \,x^{m}+c \right )^{2}}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left (x^{n} a +b \,x^{m}+c \right ) y^{\prime }-a \,x^{-2+n} y^{2}-b \,x^{m -1} y=c \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {a \,x^{-2+n} y^{2}+b \,x^{m -1} y+c}{x^{n} a +b \,x^{m}+c} \end {array} \]
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) = -(b*m*x^m-x^m*n*b-x^(m-1)*b*x+2*x^n*a+2*b*x^m-c*n+2*c)*(diff(y(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^(n-2)*a*y(x)^2/(x^n*a+b*x^m+c)+y(x)+x^(m-1)*b*y(x)*x/(x^n*a+b*x^m+c)+x^2*c/(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((a*x^n+b*x^m+c)*diff(y(x),x)=a*x^(n-2)*y(x)^2+b*x^(m-1)*y(x)+c,y(x), singsol=all)
\[ \text {No solution found} \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[(a*x^n+b*x^m+c)*y'[x]==a*x^(n-2)*y[x]^2+b*x^(m-1)*y[x]+c,y[x],x,IncludeSingularSolutions -> True]
Not solved