Internal problem ID [10475]
Internal file name [OUTPUT/9423_Monday_June_06_2022_02_31_52_PM_82473112/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 1.2.5-1. Equations Containing
Logarithmic Functions
Problem number: 1.
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]
\[ \boxed {y^{\prime }-a \ln \left (x \right )^{n} y^{2}=b m \,x^{m -1}-a \,b^{2} x^{2 m} \ln \left (x \right )^{n}} \]
In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= a \ln \left (x \right )^{n} y^{2}+b m \,x^{m -1}-a \,b^{2} x^{2 m} \ln \left (x \right )^{n} \end {align*}
This is a Riccati ODE. Comparing the ODE to solve \[ y' = -a \,b^{2} x^{2 m} \ln \left (x \right )^{n}+a \ln \left (x \right )^{n} y^{2}+\frac {b \,x^{m} m}{x} \] With Riccati ODE standard form \[ y' = f_0(x)+ f_1(x)y+f_2(x)y^{2} \] Shows that \(f_0(x)=b m \,x^{m -1}-a \,b^{2} x^{2 m} \ln \left (x \right )^{n}\), \(f_1(x)=0\) 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 &=0\\ f_2^2 f_0 &=a^{2} \ln \left (x \right )^{2 n} \left (b m \,x^{m -1}-a \,b^{2} x^{2 m} \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 )-\frac {a \ln \left (x \right )^{n} n u^{\prime }\left (x \right )}{x \ln \left (x \right )}+a^{2} \ln \left (x \right )^{2 n} \left (b m \,x^{m -1}-a \,b^{2} x^{2 m} \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
\[ u \left (x \right ) = \operatorname {DESol}\left (\left \{\textit {\_Y}^{\prime \prime }\left (x \right )-\frac {n \textit {\_Y}^{\prime }\left (x \right )}{x \ln \left (x \right )}+a \textit {\_Y} \left (x \right ) \left (\ln \left (x \right )^{n} b m \,x^{m -1}-a \,b^{2} x^{2 m} \ln \left (x \right )^{2 n}\right )\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right ) \] The above shows that \[ u^{\prime }\left (x \right ) = \frac {\partial }{\partial x}\operatorname {DESol}\left (\left \{\textit {\_Y}^{\prime \prime }\left (x \right )-\frac {n \textit {\_Y}^{\prime }\left (x \right )}{x \ln \left (x \right )}+a \textit {\_Y} \left (x \right ) \left (\ln \left (x \right )^{n} b m \,x^{m -1}-a \,b^{2} x^{2 m} \ln \left (x \right )^{2 n}\right )\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right ) \] Using the above in (1) gives the solution \[ y = -\frac {\left (\frac {\partial }{\partial x}\operatorname {DESol}\left (\left \{\textit {\_Y}^{\prime \prime }\left (x \right )-\frac {n \textit {\_Y}^{\prime }\left (x \right )}{x \ln \left (x \right )}+a \textit {\_Y} \left (x \right ) \left (\ln \left (x \right )^{n} b m \,x^{m -1}-a \,b^{2} x^{2 m} \ln \left (x \right )^{2 n}\right )\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )\right ) \ln \left (x \right )^{-n}}{a \operatorname {DESol}\left (\left \{\textit {\_Y}^{\prime \prime }\left (x \right )-\frac {n \textit {\_Y}^{\prime }\left (x \right )}{x \ln \left (x \right )}+a \textit {\_Y} \left (x \right ) \left (\ln \left (x \right )^{n} b m \,x^{m -1}-a \,b^{2} x^{2 m} \ln \left (x \right )^{2 n}\right )\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )} \] Dividing both numerator and denominator by \(c_{1}\) gives, after renaming the constant \(\frac {c_{2}}{c_{1}}=c_{3}\) the following solution
\[ y = -\frac {\left (\frac {\partial }{\partial x}\operatorname {DESol}\left (\left \{\textit {\_Y}^{\prime \prime }\left (x \right )-\frac {n \textit {\_Y}^{\prime }\left (x \right )}{x \ln \left (x \right )}+a \textit {\_Y} \left (x \right ) \left (\ln \left (x \right )^{n} b m \,x^{m -1}-a \,b^{2} x^{2 m} \ln \left (x \right )^{2 n}\right )\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )\right ) \ln \left (x \right )^{-n}}{a \operatorname {DESol}\left (\left \{\frac {-a^{2} b^{2} \textit {\_Y} \left (x \right ) \ln \left (x \right )^{1+2 n} x^{1+2 m}+a b m \,x^{m} \textit {\_Y} \left (x \right ) \ln \left (x \right )^{1+n}+\textit {\_Y}^{\prime \prime }\left (x \right ) x \ln \left (x \right )-n \textit {\_Y}^{\prime }\left (x \right )}{x \ln \left (x \right )}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )} \]
The solution(s) found are the following \begin{align*} \tag{1} y &= -\frac {\left (\frac {\partial }{\partial x}\operatorname {DESol}\left (\left \{\textit {\_Y}^{\prime \prime }\left (x \right )-\frac {n \textit {\_Y}^{\prime }\left (x \right )}{x \ln \left (x \right )}+a \textit {\_Y} \left (x \right ) \left (\ln \left (x \right )^{n} b m \,x^{m -1}-a \,b^{2} x^{2 m} \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 \{\frac {-a^{2} b^{2} \textit {\_Y} \left (x \right ) \ln \left (x \right )^{1+2 n} x^{1+2 m}+a b m \,x^{m} \textit {\_Y} \left (x \right ) \ln \left (x \right )^{1+n}+\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
\[ y = -\frac {\left (\frac {\partial }{\partial x}\operatorname {DESol}\left (\left \{\textit {\_Y}^{\prime \prime }\left (x \right )-\frac {n \textit {\_Y}^{\prime }\left (x \right )}{x \ln \left (x \right )}+a \textit {\_Y} \left (x \right ) \left (\ln \left (x \right )^{n} b m \,x^{m -1}-a \,b^{2} x^{2 m} \ln \left (x \right )^{2 n}\right )\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )\right ) \ln \left (x \right )^{-n}}{a \operatorname {DESol}\left (\left \{\frac {-a^{2} b^{2} \textit {\_Y} \left (x \right ) \ln \left (x \right )^{1+2 n} x^{1+2 m}+a b m \,x^{m} \textit {\_Y} \left (x \right ) \ln \left (x \right )^{1+n}+\textit {\_Y}^{\prime \prime }\left (x \right ) x \ln \left (x \right )-n \textit {\_Y}^{\prime }\left (x \right )}{x \ln \left (x \right )}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-a \ln \left (x \right )^{n} y^{2}=b m \,x^{m -1}-a \,b^{2} x^{2 m} \ln \left (x \right )^{n} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=a \ln \left (x \right )^{n} y^{2}+b m \,x^{m -1}-a \,b^{2} x^{2 m} \ln \left (x \right )^{n} \end {array} \]
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) = n*(diff(y(x), x))/(x*ln(x))+a*ln(x)^n*b*(ln(x)^n*x^(2*m)*a*b-x^(m-1)*m 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)+x^2*(b*m*x^(m-1)-a*b^2*x^(2*m)*ln(x)^n))/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)=a*(ln(x))^n*y(x)^2+b*m*x^(m-1)-a*b^2*x^(2*m)*(ln(x))^n,y(x), singsol=all)
\[ \text {No solution found} \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y'[x]==a*(Log[x])^n*y[x]^2+b*m*x^(m-1)-a*b^2*x^(2*m)*(Log[x])^n,y[x],x,IncludeSingularSolutions -> True]
Not solved