Internal problem ID [10490]
Internal file name [OUTPUT/9438_Monday_June_06_2022_02_32_27_PM_544547/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 1.2.5-2
Problem number: 16.
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]
\[ \boxed {y^{\prime }-a \ln \left (x \right )^{n} y^{2}-b \ln \left (x \right )^{m} y=b c \ln \left (x \right )^{m}-a \,c^{2} \ln \left (x \right )^{n}} \]
In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= a \ln \left (x \right )^{n} y^{2}+b \ln \left (x \right )^{m} y +b c \ln \left (x \right )^{m}-a \,c^{2} \ln \left (x \right )^{n} \end {align*}
This is a Riccati ODE. Comparing the ODE to solve \[ y' = a \ln \left (x \right )^{n} y^{2}+b \ln \left (x \right )^{m} y +b c \ln \left (x \right )^{m}-a \,c^{2} \ln \left (x \right )^{n} \] With Riccati ODE standard form \[ y' = f_0(x)+ f_1(x)y+f_2(x)y^{2} \] Shows that \(f_0(x)=b c \ln \left (x \right )^{m}-a \,c^{2} \ln \left (x \right )^{n}\), \(f_1(x)=\ln \left (x \right )^{m} b\) and \(f_2(x)=a \ln \left (x \right )^{n}\). Let \begin {align*} y &= \frac {-u'}{f_2 u} \\ &= \frac {-u'}{a \ln \left (x \right )^{n} 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 {a \ln \left (x \right )^{n} n}{x \ln \left (x \right )}\\ f_1 f_2 &=\ln \left (x \right )^{m} b a \ln \left (x \right )^{n}\\ f_2^2 f_0 &=a^{2} \ln \left (x \right )^{2 n} \left (b c \ln \left (x \right )^{m}-a \,c^{2} \ln \left (x \right )^{n}\right ) \end {align*}
Substituting the above terms back in equation (2) gives \begin {align*} a \ln \left (x \right )^{n} u^{\prime \prime }\left (x \right )-\left (\frac {a \ln \left (x \right )^{n} n}{x \ln \left (x \right )}+\ln \left (x \right )^{m} b a \ln \left (x \right )^{n}\right ) u^{\prime }\left (x \right )+a^{2} \ln \left (x \right )^{2 n} \left (b c \ln \left (x \right )^{m}-a \,c^{2} \ln \left (x \right )^{n}\right ) u \left (x \right ) &=0 \end {align*}
Solving the above ODE (this ode solved using Maple, not this program), gives
\[ u \left (x \right ) = \operatorname {DESol}\left (\left \{\textit {\_Y}^{\prime \prime }\left (x \right )-\textit {\_Y}^{\prime }\left (x \right ) \left (\frac {n}{x \ln \left (x \right )}+\ln \left (x \right )^{m} b \right )+a \textit {\_Y} \left (x \right ) \left (b \ln \left (x \right )^{m +n} c -a \,c^{2} \ln \left (x \right )^{2 n}\right )\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right ) \] The above shows that \[ u^{\prime }\left (x \right ) = \frac {\partial }{\partial x}\operatorname {DESol}\left (\left \{\textit {\_Y}^{\prime \prime }\left (x \right )-\textit {\_Y}^{\prime }\left (x \right ) \left (\frac {n}{x \ln \left (x \right )}+\ln \left (x \right )^{m} b \right )+a \textit {\_Y} \left (x \right ) \left (b \ln \left (x \right )^{m +n} c -a \,c^{2} \ln \left (x \right )^{2 n}\right )\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right ) \] Using the above in (1) gives the solution \[ y = -\frac {\left (\frac {\partial }{\partial x}\operatorname {DESol}\left (\left \{\textit {\_Y}^{\prime \prime }\left (x \right )-\textit {\_Y}^{\prime }\left (x \right ) \left (\frac {n}{x \ln \left (x \right )}+\ln \left (x \right )^{m} b \right )+a \textit {\_Y} \left (x \right ) \left (b \ln \left (x \right )^{m +n} c -a \,c^{2} \ln \left (x \right )^{2 n}\right )\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )\right ) \ln \left (x \right )^{-n}}{a \operatorname {DESol}\left (\left \{\textit {\_Y}^{\prime \prime }\left (x \right )-\textit {\_Y}^{\prime }\left (x \right ) \left (\frac {n}{x \ln \left (x \right )}+\ln \left (x \right )^{m} b \right )+a \textit {\_Y} \left (x \right ) \left (b \ln \left (x \right )^{m +n} c -a \,c^{2} \ln \left (x \right )^{2 n}\right )\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )} \] Dividing both numerator and denominator by \(c_{1}\) gives, after renaming the constant \(\frac {c_{2}}{c_{1}}=c_{3}\) the following solution
\[ y = -\frac {\left (\frac {\partial }{\partial x}\operatorname {DESol}\left (\left \{\frac {-\ln \left (x \right ) \ln \left (x \right )^{2 n} \textit {\_Y} \left (x \right ) a^{2} c^{2} x +\ln \left (x \right ) \ln \left (x \right )^{m +n} \textit {\_Y} \left (x \right ) a b c x -\textit {\_Y}^{\prime }\left (x \right ) \ln \left (x \right )^{m +1} b x +\textit {\_Y}^{\prime \prime }\left (x \right ) x \ln \left (x \right )-n \textit {\_Y}^{\prime }\left (x \right )}{x \ln \left (x \right )}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )\right ) \ln \left (x \right )^{-n}}{a \operatorname {DESol}\left (\left \{\frac {-x \,a^{2} c^{2} \textit {\_Y} \left (x \right ) \ln \left (x \right )^{1+2 n}+a b c x \textit {\_Y} \left (x \right ) \ln \left (x \right )^{1+m +n}-\textit {\_Y}^{\prime }\left (x \right ) \ln \left (x \right )^{m +1} b x +\textit {\_Y}^{\prime \prime }\left (x \right ) x \ln \left (x \right )-n \textit {\_Y}^{\prime }\left (x \right )}{x \ln \left (x \right )}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )} \]
The solution(s) found are the following \begin{align*} \tag{1} y &= -\frac {\left (\frac {\partial }{\partial x}\operatorname {DESol}\left (\left \{\frac {-\ln \left (x \right ) \ln \left (x \right )^{2 n} \textit {\_Y} \left (x \right ) a^{2} c^{2} x +\ln \left (x \right ) \ln \left (x \right )^{m +n} \textit {\_Y} \left (x \right ) a b c x -\textit {\_Y}^{\prime }\left (x \right ) \ln \left (x \right )^{m +1} b x +\textit {\_Y}^{\prime \prime }\left (x \right ) x \ln \left (x \right )-n \textit {\_Y}^{\prime }\left (x \right )}{x \ln \left (x \right )}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )\right ) \ln \left (x \right )^{-n}}{a \operatorname {DESol}\left (\left \{\frac {-x \,a^{2} c^{2} \textit {\_Y} \left (x \right ) \ln \left (x \right )^{1+2 n}+a b c x \textit {\_Y} \left (x \right ) \ln \left (x \right )^{1+m +n}-\textit {\_Y}^{\prime }\left (x \right ) \ln \left (x \right )^{m +1} b x +\textit {\_Y}^{\prime \prime }\left (x \right ) x \ln \left (x \right )-n \textit {\_Y}^{\prime }\left (x \right )}{x \ln \left (x \right )}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )} \\ \end{align*}
Verification of solutions
\[ y = -\frac {\left (\frac {\partial }{\partial x}\operatorname {DESol}\left (\left \{\frac {-\ln \left (x \right ) \ln \left (x \right )^{2 n} \textit {\_Y} \left (x \right ) a^{2} c^{2} x +\ln \left (x \right ) \ln \left (x \right )^{m +n} \textit {\_Y} \left (x \right ) a b c x -\textit {\_Y}^{\prime }\left (x \right ) \ln \left (x \right )^{m +1} b x +\textit {\_Y}^{\prime \prime }\left (x \right ) x \ln \left (x \right )-n \textit {\_Y}^{\prime }\left (x \right )}{x \ln \left (x \right )}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )\right ) \ln \left (x \right )^{-n}}{a \operatorname {DESol}\left (\left \{\frac {-x \,a^{2} c^{2} \textit {\_Y} \left (x \right ) \ln \left (x \right )^{1+2 n}+a b c x \textit {\_Y} \left (x \right ) \ln \left (x \right )^{1+m +n}-\textit {\_Y}^{\prime }\left (x \right ) \ln \left (x \right )^{m +1} b x +\textit {\_Y}^{\prime \prime }\left (x \right ) x \ln \left (x \right )-n \textit {\_Y}^{\prime }\left (x \right )}{x \ln \left (x \right )}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-a \ln \left (x \right )^{n} y^{2}-b \ln \left (x \right )^{m} y=b c \ln \left (x \right )^{m}-a \,c^{2} \ln \left (x \right )^{n} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=a \ln \left (x \right )^{n} y^{2}+b \ln \left (x \right )^{m} y+b c \ln \left (x \right )^{m}-a \,c^{2} \ln \left (x \right )^{n} \end {array} \]
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) = (ln(x)^m*ln(x)*b*x+n)*(diff(y(x), x))/(x*ln(x))-ln(x)^n*a*c*(-ln(x)^n* 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 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 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 <- 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 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 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 <- 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*ln(x)^n*y(x)^2+y(x)+ln(x)^m*b*y(x)*x+x^2*(b*c*ln(x)^m-a*c^2*ln(x)^n))/x, y(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)] <- symmetry pattern of the form [0, F(x)*G(y)] successful <- Riccati with symmetry pattern of the form [0,F(x)*G(y)] successful`
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 99
dsolve(diff(y(x),x)=a*(ln(x))^n*y(x)^2+b*(ln(x))^m*y(x)+b*c*(ln(x))^m-a*c^2*(ln(x))^n,y(x), singsol=all)
\[ y \left (x \right ) = \frac {-c a \left (\int \ln \left (x \right )^{n} {\mathrm e}^{-\left (\int \left (2 \ln \left (x \right )^{n} a c -\ln \left (x \right )^{m} b \right )d x \right )}d x \right )-c_{1} c -{\mathrm e}^{-\left (\int \left (2 \ln \left (x \right )^{n} a c -\ln \left (x \right )^{m} b \right )d x \right )}}{c_{1} +a \left (\int \ln \left (x \right )^{n} {\mathrm e}^{-\left (\int \left (2 \ln \left (x \right )^{n} a c -\ln \left (x \right )^{m} b \right )d x \right )}d x \right )} \]
✓ Solution by Mathematica
Time used: 3.927 (sec). Leaf size: 385
DSolve[y'[x]==a*(Log[x])^n*y[x]^2+b*(Log[x])^m*y[x]+b*c*(Log[x])^m-a*c^2*(Log[x])^n,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^x\frac {\exp \left (b \Gamma (m+1,-\log (K[1])) (-\log (K[1]))^{-m} \log ^m(K[1])-2 a c \Gamma (n+1,-\log (K[1])) (-\log (K[1]))^{-n} \log ^n(K[1])\right ) \left (-b \log ^m(K[1])+a c \log ^n(K[1])-a y(x) \log ^n(K[1])\right )}{a b (m-n) (c+y(x))}dK[1]+\int _1^{y(x)}\left (\frac {\exp \left (b \Gamma (m+1,-\log (x)) (-\log (x))^{-m} \log ^m(x)-2 a c \Gamma (n+1,-\log (x)) (-\log (x))^{-n} \log ^n(x)\right )}{a b (m-n) (c+K[2])^2}-\int _1^x\left (-\frac {\exp \left (b \Gamma (m+1,-\log (K[1])) (-\log (K[1]))^{-m} \log ^m(K[1])-2 a c \Gamma (n+1,-\log (K[1])) (-\log (K[1]))^{-n} \log ^n(K[1])\right ) \log ^n(K[1])}{b (m-n) (c+K[2])}-\frac {\exp \left (b \Gamma (m+1,-\log (K[1])) (-\log (K[1]))^{-m} \log ^m(K[1])-2 a c \Gamma (n+1,-\log (K[1])) (-\log (K[1]))^{-n} \log ^n(K[1])\right ) \left (-b \log ^m(K[1])+a c \log ^n(K[1])-a K[2] \log ^n(K[1])\right )}{a b (m-n) (c+K[2])^2}\right )dK[1]\right )dK[2]=c_1,y(x)\right ] \]