Internal problem ID [10352]
Internal file name [OUTPUT/9300_Monday_June_06_2022_01_49_33_PM_14490210/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: 23.
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]
\[ \boxed {\left (x^{n} a +b \,x^{m}+c \right ) \left (y^{\prime }-y^{2}\right )=-a n \left (-1+n \right ) x^{-2+n}-b m \left (m -1\right ) x^{m -2}} \]
In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= \frac {x^{m} y^{2} b -a \,n^{2} x^{-2+n}-b \,m^{2} x^{m -2}+a \,x^{n} y^{2}+a n \,x^{-2+n}+b m \,x^{m -2}+c \,y^{2}}{x^{n} a +b \,x^{m}+c} \end {align*}
This is a Riccati ODE. Comparing the ODE to solve \[ y' = \frac {x^{m} y^{2} b}{x^{n} a +b \,x^{m}+c}-\frac {a \,n^{2} x^{n}}{\left (x^{n} a +b \,x^{m}+c \right ) x^{2}}-\frac {b \,m^{2} x^{m}}{\left (x^{n} a +b \,x^{m}+c \right ) x^{2}}+\frac {a \,x^{n} y^{2}}{x^{n} a +b \,x^{m}+c}+\frac {a n \,x^{n}}{\left (x^{n} a +b \,x^{m}+c \right ) x^{2}}+\frac {b m \,x^{m}}{\left (x^{n} a +b \,x^{m}+c \right ) x^{2}}+\frac {c \,y^{2}}{x^{n} a +b \,x^{m}+c} \] With Riccati ODE standard form \[ y' = f_0(x)+ f_1(x)y+f_2(x)y^{2} \] Shows that \(f_0(x)=\frac {-a \,n^{2} x^{-2+n}-b \,m^{2} x^{m -2}+a n \,x^{-2+n}+b m \,x^{m -2}}{x^{n} a +b \,x^{m}+c}\), \(f_1(x)=0\) and \(f_2(x)=1\). Let \begin {align*} y &= \frac {-u'}{f_2 u} \\ &= \frac {-u'}{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' &=0\\ f_1 f_2 &=0\\ f_2^2 f_0 &=\frac {-a \,n^{2} x^{-2+n}-b \,m^{2} x^{m -2}+a n \,x^{-2+n}+b m \,x^{m -2}}{x^{n} a +b \,x^{m}+c} \end {align*}
Substituting the above terms back in equation (2) gives \begin {align*} u^{\prime \prime }\left (x \right )+\frac {\left (-a \,n^{2} x^{-2+n}-b \,m^{2} x^{m -2}+a n \,x^{-2+n}+b m \,x^{m -2}\right ) u \left (x \right )}{x^{n} a +b \,x^{m}+c} &=0 \end {align*}
Solving the above ODE (this ode solved using Maple, not this program), gives
\[ u \left (x \right ) = \left (\left (\int \frac {1}{\left (x^{n} a +b \,x^{m}+c \right )^{2}}d x \right ) c_{1} +c_{2} \right ) \left (x^{n} a +b \,x^{m}+c \right ) \] The above shows that \[ u^{\prime }\left (x \right ) = \frac {c_{1}}{x^{n} a +b \,x^{m}+c}+\frac {\left (\left (\int \frac {1}{\left (x^{n} a +b \,x^{m}+c \right )^{2}}d x \right ) c_{1} +c_{2} \right ) \left (x^{n} n a +b \,x^{m} m \right )}{x} \] Using the above in (1) gives the solution \[ y = -\frac {\frac {c_{1}}{x^{n} a +b \,x^{m}+c}+\frac {\left (\left (\int \frac {1}{\left (x^{n} a +b \,x^{m}+c \right )^{2}}d x \right ) c_{1} +c_{2} \right ) \left (x^{n} n a +b \,x^{m} m \right )}{x}}{\left (\left (\int \frac {1}{\left (x^{n} a +b \,x^{m}+c \right )^{2}}d x \right ) c_{1} +c_{2} \right ) \left (x^{n} a +b \,x^{m}+c \right )} \] Dividing both numerator and denominator by \(c_{1}\) gives, after renaming the constant \(\frac {c_{2}}{c_{1}}=c_{3}\) the following solution
\[ y = \frac {-\frac {c_{3}}{x^{n} a +b \,x^{m}+c}-\frac {\left (\left (\int \frac {1}{\left (x^{n} a +b \,x^{m}+c \right )^{2}}d x \right ) c_{3} +1\right ) \left (x^{n} n a +b \,x^{m} m \right )}{x}}{\left (\left (\int \frac {1}{\left (x^{n} a +b \,x^{m}+c \right )^{2}}d x \right ) c_{3} +1\right ) \left (x^{n} a +b \,x^{m}+c \right )} \]
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {-\frac {c_{3}}{x^{n} a +b \,x^{m}+c}-\frac {\left (\left (\int \frac {1}{\left (x^{n} a +b \,x^{m}+c \right )^{2}}d x \right ) c_{3} +1\right ) \left (x^{n} n a +b \,x^{m} m \right )}{x}}{\left (\left (\int \frac {1}{\left (x^{n} a +b \,x^{m}+c \right )^{2}}d x \right ) c_{3} +1\right ) \left (x^{n} a +b \,x^{m}+c \right )} \\ \end{align*}
Verification of solutions
\[ y = \frac {-\frac {c_{3}}{x^{n} a +b \,x^{m}+c}-\frac {\left (\left (\int \frac {1}{\left (x^{n} a +b \,x^{m}+c \right )^{2}}d x \right ) c_{3} +1\right ) \left (x^{n} n a +b \,x^{m} m \right )}{x}}{\left (\left (\int \frac {1}{\left (x^{n} a +b \,x^{m}+c \right )^{2}}d x \right ) c_{3} +1\right ) \left (x^{n} a +b \,x^{m}+c \right )} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left (x^{n} a +b \,x^{m}+c \right ) \left (y^{\prime }-y^{2}\right )=-a n \left (-1+n \right ) x^{-2+n}-b m \left (m -1\right ) x^{m -2} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {x^{m} y^{2} b -a \,n^{2} x^{-2+n}-b \,m^{2} x^{m -2}+a \,x^{n} y^{2}+a n \,x^{-2+n}+b m \,x^{m -2}+c y^{2}}{x^{n} a +b \,x^{m}+c} \end {array} \]
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^2*x^(-2+m)+a*n^2*x^(n-2)-b*m*x^(-2+m)-a*n*x^(n-2))*y(x)/(x^n*a+b* Methods for second order ODEs: --- Trying classification methods --- trying a symmetry of the form [xi=0, eta=F(x)] <- linear_1 successful <- Riccati to 2nd Order successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 141
dsolve((a*x^n+b*x^m+c)*(diff(y(x),x)-y(x)^2)+a*n*(n-1)*x^(n-2)+b*m*(m-1)*x^(m-2)=0,y(x), singsol=all)
\[ y \left (x \right ) = \frac {-\left (a n \,x^{n}+b m \,x^{m}\right ) \left (a \,x^{n}+b \,x^{m}+c \right ) \left (\int \frac {1}{\left (a \,x^{n}+b \,x^{m}+c \right )^{2}}d x \right )-x^{2 m} c_{1} b^{2} m -b \left (a \left (n +m \right ) x^{n}+c m \right ) c_{1} x^{m}-x^{2 n} c_{1} a^{2} n -x^{n} c_{1} a c n -x}{\left (a \,x^{n}+b \,x^{m}+c \right )^{2} x \left (c_{1} +\int \frac {1}{\left (a \,x^{n}+b \,x^{m}+c \right )^{2}}d x \right )} \]
✓ Solution by Mathematica
Time used: 35.099 (sec). Leaf size: 201
DSolve[(a*x^n+b*x^m+c)*(y'[x]-y[x]^2)+a*n*(n-1)*x^(n-2)+b*m*(m-1)*x^(m-2)==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {c_1 \left (\frac {\left (a n x^n+b m x^m\right ) \int _1^x\frac {1}{\left (b K[1]^m+a K[1]^n+c\right )^2}dK[1]}{x}+\frac {1}{a x^n+b x^m+c}\right )+a n x^{n-1}+b m x^{m-1}}{\left (a x^n+b x^m+c\right ) \left (1+c_1 \int _1^x\frac {1}{\left (b K[1]^m+a K[1]^n+c\right )^2}dK[1]\right )} \\ y(x)\to -\frac {\frac {1}{\int _1^x\frac {1}{\left (b K[1]^m+a K[1]^n+c\right )^2}dK[1]}+\frac {\left (a n x^n+b m x^m\right ) \left (a x^n+b x^m+c\right )}{x}}{\left (a x^n+b x^m+c\right )^2} \\ \end{align*}