Internal problem ID [10631]
Internal file name [OUTPUT/9579_Monday_June_06_2022_03_10_49_PM_7500533/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 1.2.8-2. Equations containing
arbitrary functions and their derivatives.
Problem number: 39.
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]
\[ \boxed {f \left (x \right )^{2} y^{\prime }-f^{\prime }\left (x \right ) y^{2}+g \left (x \right ) \left (y-f \left (x \right )\right )=0} \]
In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= \frac {f^{\prime }\left (x \right ) y^{2}+f \left (x \right ) g \left (x \right )-g \left (x \right ) y}{f \left (x \right )^{2}} \end {align*}
This is a Riccati ODE. Comparing the ODE to solve \[ y' = \frac {f^{\prime }\left (x \right ) y^{2}}{f \left (x \right )^{2}}+\frac {g \left (x \right )}{f \left (x \right )}-\frac {g \left (x \right ) y}{f \left (x \right )^{2}} \] With Riccati ODE standard form \[ y' = f_0(x)+ f_1(x)y+f_2(x)y^{2} \] Shows that \(f_0(x)=\frac {g \left (x \right )}{f \left (x \right )}\), \(f_1(x)=-\frac {g \left (x \right )}{f \left (x \right )^{2}}\) and \(f_2(x)=\frac {f^{\prime }\left (x \right )}{f \left (x \right )^{2}}\). Let \begin {align*} y &= \frac {-u'}{f_2 u} \\ &= \frac {-u'}{\frac {f^{\prime }\left (x \right ) u}{f \left (x \right )^{2}}} \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 {2 {f^{\prime }\left (x \right )}^{2}}{f \left (x \right )^{3}}+\frac {f^{\prime \prime }\left (x \right )}{f \left (x \right )^{2}}\\ f_1 f_2 &=-\frac {g \left (x \right ) f^{\prime }\left (x \right )}{f \left (x \right )^{4}}\\ f_2^2 f_0 &=\frac {{f^{\prime }\left (x \right )}^{2} g \left (x \right )}{f \left (x \right )^{5}} \end {align*}
Substituting the above terms back in equation (2) gives \begin {align*} \frac {f^{\prime }\left (x \right ) u^{\prime \prime }\left (x \right )}{f \left (x \right )^{2}}-\left (-\frac {2 {f^{\prime }\left (x \right )}^{2}}{f \left (x \right )^{3}}+\frac {f^{\prime \prime }\left (x \right )}{f \left (x \right )^{2}}-\frac {g \left (x \right ) f^{\prime }\left (x \right )}{f \left (x \right )^{4}}\right ) u^{\prime }\left (x \right )+\frac {{f^{\prime }\left (x \right )}^{2} g \left (x \right ) u \left (x \right )}{f \left (x \right )^{5}} &=0 \end {align*}
Solving the above ODE (this ode solved using Maple, not this program), gives
\[ u \left (x \right ) = \operatorname {DESol}\left (\left \{\frac {2 {f^{\prime }\left (x \right )}^{2} f \left (x \right )^{2} \textit {\_Y}^{\prime }\left (x \right )+\textit {\_Y}^{\prime \prime }\left (x \right ) f \left (x \right )^{3} f^{\prime }\left (x \right )-f \left (x \right )^{3} \textit {\_Y}^{\prime }\left (x \right ) f^{\prime \prime }\left (x \right )+{f^{\prime }\left (x \right )}^{2} g \left (x \right ) \textit {\_Y} \left (x \right )+f^{\prime }\left (x \right ) f \left (x \right ) g \left (x \right ) \textit {\_Y}^{\prime }\left (x \right )}{f \left (x \right )^{3} f^{\prime }\left (x \right )}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right ) \] The above shows that \[ u^{\prime }\left (x \right ) = \frac {d}{d x}\operatorname {DESol}\left (\left \{\frac {2 {f^{\prime }\left (x \right )}^{2} f \left (x \right )^{2} \textit {\_Y}^{\prime }\left (x \right )+\textit {\_Y}^{\prime \prime }\left (x \right ) f \left (x \right )^{3} f^{\prime }\left (x \right )-f \left (x \right )^{3} \textit {\_Y}^{\prime }\left (x \right ) f^{\prime \prime }\left (x \right )+{f^{\prime }\left (x \right )}^{2} g \left (x \right ) \textit {\_Y} \left (x \right )+f^{\prime }\left (x \right ) f \left (x \right ) g \left (x \right ) \textit {\_Y}^{\prime }\left (x \right )}{f \left (x \right )^{3} f^{\prime }\left (x \right )}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right ) \] Using the above in (1) gives the solution \[ y = -\frac {\left (\frac {d}{d x}\operatorname {DESol}\left (\left \{\frac {2 {f^{\prime }\left (x \right )}^{2} f \left (x \right )^{2} \textit {\_Y}^{\prime }\left (x \right )+\textit {\_Y}^{\prime \prime }\left (x \right ) f \left (x \right )^{3} f^{\prime }\left (x \right )-f \left (x \right )^{3} \textit {\_Y}^{\prime }\left (x \right ) f^{\prime \prime }\left (x \right )+{f^{\prime }\left (x \right )}^{2} g \left (x \right ) \textit {\_Y} \left (x \right )+f^{\prime }\left (x \right ) f \left (x \right ) g \left (x \right ) \textit {\_Y}^{\prime }\left (x \right )}{f \left (x \right )^{3} f^{\prime }\left (x \right )}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )\right ) f \left (x \right )^{2}}{f^{\prime }\left (x \right ) \operatorname {DESol}\left (\left \{\frac {2 {f^{\prime }\left (x \right )}^{2} f \left (x \right )^{2} \textit {\_Y}^{\prime }\left (x \right )+\textit {\_Y}^{\prime \prime }\left (x \right ) f \left (x \right )^{3} f^{\prime }\left (x \right )-f \left (x \right )^{3} \textit {\_Y}^{\prime }\left (x \right ) f^{\prime \prime }\left (x \right )+{f^{\prime }\left (x \right )}^{2} g \left (x \right ) \textit {\_Y} \left (x \right )+f^{\prime }\left (x \right ) f \left (x \right ) g \left (x \right ) \textit {\_Y}^{\prime }\left (x \right )}{f \left (x \right )^{3} f^{\prime }\left (x \right )}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )} \] Dividing both numerator and denominator by \(c_{1}\) gives, after renaming the constant \(\frac {c_{2}}{c_{1}}=c_{3}\) the following solution
\[ y = -\frac {\left (\frac {d}{d x}\operatorname {DESol}\left (\left \{\frac {2 {f^{\prime }\left (x \right )}^{2} f \left (x \right )^{2} \textit {\_Y}^{\prime }\left (x \right )+\textit {\_Y}^{\prime \prime }\left (x \right ) f \left (x \right )^{3} f^{\prime }\left (x \right )-f \left (x \right )^{3} \textit {\_Y}^{\prime }\left (x \right ) f^{\prime \prime }\left (x \right )+{f^{\prime }\left (x \right )}^{2} g \left (x \right ) \textit {\_Y} \left (x \right )+f^{\prime }\left (x \right ) f \left (x \right ) g \left (x \right ) \textit {\_Y}^{\prime }\left (x \right )}{f \left (x \right )^{3} f^{\prime }\left (x \right )}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )\right ) f \left (x \right )^{2}}{f^{\prime }\left (x \right ) \operatorname {DESol}\left (\left \{\frac {2 {f^{\prime }\left (x \right )}^{2} f \left (x \right )^{2} \textit {\_Y}^{\prime }\left (x \right )+\textit {\_Y}^{\prime \prime }\left (x \right ) f \left (x \right )^{3} f^{\prime }\left (x \right )-f \left (x \right )^{3} \textit {\_Y}^{\prime }\left (x \right ) f^{\prime \prime }\left (x \right )+{f^{\prime }\left (x \right )}^{2} g \left (x \right ) \textit {\_Y} \left (x \right )+f^{\prime }\left (x \right ) f \left (x \right ) g \left (x \right ) \textit {\_Y}^{\prime }\left (x \right )}{f \left (x \right )^{3} f^{\prime }\left (x \right )}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )} \]
The solution(s) found are the following \begin{align*} \tag{1} y &= -\frac {\left (\frac {d}{d x}\operatorname {DESol}\left (\left \{\frac {2 {f^{\prime }\left (x \right )}^{2} f \left (x \right )^{2} \textit {\_Y}^{\prime }\left (x \right )+\textit {\_Y}^{\prime \prime }\left (x \right ) f \left (x \right )^{3} f^{\prime }\left (x \right )-f \left (x \right )^{3} \textit {\_Y}^{\prime }\left (x \right ) f^{\prime \prime }\left (x \right )+{f^{\prime }\left (x \right )}^{2} g \left (x \right ) \textit {\_Y} \left (x \right )+f^{\prime }\left (x \right ) f \left (x \right ) g \left (x \right ) \textit {\_Y}^{\prime }\left (x \right )}{f \left (x \right )^{3} f^{\prime }\left (x \right )}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )\right ) f \left (x \right )^{2}}{f^{\prime }\left (x \right ) \operatorname {DESol}\left (\left \{\frac {2 {f^{\prime }\left (x \right )}^{2} f \left (x \right )^{2} \textit {\_Y}^{\prime }\left (x \right )+\textit {\_Y}^{\prime \prime }\left (x \right ) f \left (x \right )^{3} f^{\prime }\left (x \right )-f \left (x \right )^{3} \textit {\_Y}^{\prime }\left (x \right ) f^{\prime \prime }\left (x \right )+{f^{\prime }\left (x \right )}^{2} g \left (x \right ) \textit {\_Y} \left (x \right )+f^{\prime }\left (x \right ) f \left (x \right ) g \left (x \right ) \textit {\_Y}^{\prime }\left (x \right )}{f \left (x \right )^{3} f^{\prime }\left (x \right )}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )} \\ \end{align*}
Verification of solutions
\[ y = -\frac {\left (\frac {d}{d x}\operatorname {DESol}\left (\left \{\frac {2 {f^{\prime }\left (x \right )}^{2} f \left (x \right )^{2} \textit {\_Y}^{\prime }\left (x \right )+\textit {\_Y}^{\prime \prime }\left (x \right ) f \left (x \right )^{3} f^{\prime }\left (x \right )-f \left (x \right )^{3} \textit {\_Y}^{\prime }\left (x \right ) f^{\prime \prime }\left (x \right )+{f^{\prime }\left (x \right )}^{2} g \left (x \right ) \textit {\_Y} \left (x \right )+f^{\prime }\left (x \right ) f \left (x \right ) g \left (x \right ) \textit {\_Y}^{\prime }\left (x \right )}{f \left (x \right )^{3} f^{\prime }\left (x \right )}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )\right ) f \left (x \right )^{2}}{f^{\prime }\left (x \right ) \operatorname {DESol}\left (\left \{\frac {2 {f^{\prime }\left (x \right )}^{2} f \left (x \right )^{2} \textit {\_Y}^{\prime }\left (x \right )+\textit {\_Y}^{\prime \prime }\left (x \right ) f \left (x \right )^{3} f^{\prime }\left (x \right )-f \left (x \right )^{3} \textit {\_Y}^{\prime }\left (x \right ) f^{\prime \prime }\left (x \right )+{f^{\prime }\left (x \right )}^{2} g \left (x \right ) \textit {\_Y} \left (x \right )+f^{\prime }\left (x \right ) f \left (x \right ) g \left (x \right ) \textit {\_Y}^{\prime }\left (x \right )}{f \left (x \right )^{3} f^{\prime }\left (x \right )}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & f \left (x \right )^{2} y^{\prime }-f^{\prime }\left (x \right ) y^{2}+g \left (x \right ) \left (y-f \left (x \right )\right )=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {f^{\prime }\left (x \right ) y^{2}-g \left (x \right ) \left (y-f \left (x \right )\right )}{f \left (x \right )^{2}} \end {array} \]
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) = -(-(diff(diff(f(x), x), x))*f(x)^2+2*f(x)*(diff(f(x), x))^2+(diff(f(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 -> 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)-((diff(f(x), x))*y(x)^2/f(x)^2+y(x)-g(x)*y(x)*x/f(x)^2+x^2*g(x)/f(x))/x, y(x), ex 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(f(x)^2*diff(y(x),x)-diff(f(x),x)*y(x)^2+g(x)*(y(x)-f(x))=0,y(x), singsol=all)
\[ \text {No solution found} \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[f[x]^2*y'[x]-f'[x]*y[x]^2+g[x]*(y[x]-f[x])==0,y[x],x,IncludeSingularSolutions -> True]
Not solved