Internal problem ID [10596]
Internal file name [OUTPUT/9544_Monday_June_06_2022_03_07_20_PM_89866051/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. subsection 1.2.8-1. Equations containing
arbitrary functions (but not containing their derivatives).
Problem number: 4.
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]
\[ \boxed {y^{\prime }-f \left (x \right ) y^{2}+a \,x^{n} f \left (x \right ) y=a n \,x^{n -1}} \]
In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= f \left (x \right ) y^{2}-a \,x^{n} f \left (x \right ) y +a n \,x^{n -1} \end {align*}
This is a Riccati ODE. Comparing the ODE to solve \[ y' = f \left (x \right ) y^{2}-a \,x^{n} f \left (x \right ) y +\frac {a n \,x^{n}}{x} \] With Riccati ODE standard form \[ y' = f_0(x)+ f_1(x)y+f_2(x)y^{2} \] Shows that \(f_0(x)=a n \,x^{n -1}\), \(f_1(x)=-a \,x^{n} f \left (x \right )\) and \(f_2(x)=f \left (x \right )\). Let \begin {align*} y &= \frac {-u'}{f_2 u} \\ &= \frac {-u'}{f \left (x \right ) 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' &=f^{\prime }\left (x \right )\\ f_1 f_2 &=-f \left (x \right )^{2} x^{n} a\\ f_2^2 f_0 &=f \left (x \right )^{2} a n \,x^{n -1} \end {align*}
Substituting the above terms back in equation (2) gives \begin {align*} f \left (x \right ) u^{\prime \prime }\left (x \right )-\left (-f \left (x \right )^{2} x^{n} a +f^{\prime }\left (x \right )\right ) u^{\prime }\left (x \right )+f \left (x \right )^{2} a n \,x^{n -1} u \left (x \right ) &=0 \end {align*}
Solving the above ODE (this ode solved using Maple, not this program), gives
\[ u \left (x \right ) = {\mathrm e}^{-a \left (\int f \left (x \right ) x^{n}d x \right )} \left (c_{1} +\left (\int f \left (x \right ) {\mathrm e}^{a \left (\int f \left (x \right ) x^{n}d x \right )}d x \right ) c_{2} \right ) \] The above shows that \[ u^{\prime }\left (x \right ) = f \left (x \right ) \left (-x^{n} {\mathrm e}^{-a \left (\int f \left (x \right ) x^{n}d x \right )} \left (\int f \left (x \right ) {\mathrm e}^{a \left (\int f \left (x \right ) x^{n}d x \right )}d x \right ) c_{2} a -x^{n} {\mathrm e}^{-a \left (\int f \left (x \right ) x^{n}d x \right )} c_{1} a +c_{2} \right ) \] Using the above in (1) gives the solution \[ y = -\frac {\left (-x^{n} {\mathrm e}^{-a \left (\int f \left (x \right ) x^{n}d x \right )} \left (\int f \left (x \right ) {\mathrm e}^{a \left (\int f \left (x \right ) x^{n}d x \right )}d x \right ) c_{2} a -x^{n} {\mathrm e}^{-a \left (\int f \left (x \right ) x^{n}d x \right )} c_{1} a +c_{2} \right ) {\mathrm e}^{\int a \,x^{n} f \left (x \right )d x}}{c_{1} +\left (\int f \left (x \right ) {\mathrm e}^{a \left (\int f \left (x \right ) x^{n}d x \right )}d x \right ) c_{2}} \] Dividing both numerator and denominator by \(c_{1}\) gives, after renaming the constant \(\frac {c_{2}}{c_{1}}=c_{3}\) the following solution
\[ y = \frac {a c_{3} x^{n}+a \left (\int f \left (x \right ) {\mathrm e}^{a \left (\int f \left (x \right ) x^{n}d x \right )}d x \right ) x^{n}-{\mathrm e}^{a \left (\int f \left (x \right ) x^{n}d x \right )}}{c_{3} +\int f \left (x \right ) {\mathrm e}^{a \left (\int f \left (x \right ) x^{n}d x \right )}d x} \]
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {a c_{3} x^{n}+a \left (\int f \left (x \right ) {\mathrm e}^{a \left (\int f \left (x \right ) x^{n}d x \right )}d x \right ) x^{n}-{\mathrm e}^{a \left (\int f \left (x \right ) x^{n}d x \right )}}{c_{3} +\int f \left (x \right ) {\mathrm e}^{a \left (\int f \left (x \right ) x^{n}d x \right )}d x} \\ \end{align*}
Verification of solutions
\[ y = \frac {a c_{3} x^{n}+a \left (\int f \left (x \right ) {\mathrm e}^{a \left (\int f \left (x \right ) x^{n}d x \right )}d x \right ) x^{n}-{\mathrm e}^{a \left (\int f \left (x \right ) x^{n}d x \right )}}{c_{3} +\int f \left (x \right ) {\mathrm e}^{a \left (\int f \left (x \right ) x^{n}d x \right )}d x} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-f \left (x \right ) y^{2}+a \,x^{n} f \left (x \right ) y=a n \,x^{n -1} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=f \left (x \right ) y^{2}-a \,x^{n} f \left (x \right ) y+a n \,x^{n -1} \end {array} \]
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*f(x)^2-(diff(f(x), x)))*(diff(y(x), x))/f(x)-f(x)*a*n*x^(n-1)* 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 -> 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 -> 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)-(f(x)*y(x)^2+y(x)-a*x^n*f(x)*y(x)*x+x^2*a*n*x^(n-1))/x, y(x), explicit` *** 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)=f(x)*y(x)^2-a*x^n*f(x)*y(x)+a*n*x^(n-1),y(x), singsol=all)
\[ \text {No solution found} \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y'[x]==f[x]*y[x]^2-a*x^n*f[x]*y[x]+a*n*x^(n-1),y[x],x,IncludeSingularSolutions -> True]
Not solved