Internal problem ID [10402]
Internal file name [OUTPUT/9350_Monday_June_06_2022_02_14_58_PM_25408529/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: 73.
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]
\[ \boxed {x \left (a \,x^{k}+b \right ) y^{\prime }-\alpha \,x^{n} y^{2}-\left (\beta -a n \,x^{k}\right ) y=\gamma \,x^{-n}} \]
In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= \frac {-x^{k} a n y +\alpha \,x^{n} y^{2}+\gamma \,x^{-n}+\beta y}{x \left (a \,x^{k}+b \right )} \end {align*}
This is a Riccati ODE. Comparing the ODE to solve \[ y' = -\frac {x^{k} a n y}{x \left (a \,x^{k}+b \right )}+\frac {\alpha \,x^{n} y^{2}}{x \left (a \,x^{k}+b \right )}+\frac {\gamma \,x^{-n}}{x \left (a \,x^{k}+b \right )}+\frac {\beta y}{x \left (a \,x^{k}+b \right )} \] With Riccati ODE standard form \[ y' = f_0(x)+ f_1(x)y+f_2(x)y^{2} \] Shows that \(f_0(x)=\frac {\gamma \,x^{-n}}{x \left (a \,x^{k}+b \right )}\), \(f_1(x)=\frac {\beta -a n \,x^{k}}{x \left (a \,x^{k}+b \right )}\) and \(f_2(x)=\frac {\alpha \,x^{n}}{x \left (a \,x^{k}+b \right )}\). Let \begin {align*} y &= \frac {-u'}{f_2 u} \\ &= \frac {-u'}{\frac {\alpha \,x^{n} u}{x \left (a \,x^{k}+b \right )}} \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 {\alpha \,x^{n} n}{x^{2} \left (a \,x^{k}+b \right )}-\frac {\alpha \,x^{n}}{x^{2} \left (a \,x^{k}+b \right )}-\frac {\alpha \,x^{n} a k \,x^{k}}{x^{2} \left (a \,x^{k}+b \right )^{2}}\\ f_1 f_2 &=\frac {\left (\beta -a n \,x^{k}\right ) \alpha \,x^{n}}{x^{2} \left (a \,x^{k}+b \right )^{2}}\\ f_2^2 f_0 &=\frac {\alpha ^{2} x^{2 n} \gamma \,x^{-n}}{x^{3} \left (a \,x^{k}+b \right )^{3}} \end {align*}
Substituting the above terms back in equation (2) gives \begin {align*} \frac {\alpha \,x^{n} u^{\prime \prime }\left (x \right )}{x \left (a \,x^{k}+b \right )}-\left (\frac {\alpha \,x^{n} n}{x^{2} \left (a \,x^{k}+b \right )}-\frac {\alpha \,x^{n}}{x^{2} \left (a \,x^{k}+b \right )}-\frac {\alpha \,x^{n} a k \,x^{k}}{x^{2} \left (a \,x^{k}+b \right )^{2}}+\frac {\left (\beta -a n \,x^{k}\right ) \alpha \,x^{n}}{x^{2} \left (a \,x^{k}+b \right )^{2}}\right ) u^{\prime }\left (x \right )+\frac {\alpha ^{2} x^{2 n} \gamma \,x^{-n} u \left (x \right )}{x^{3} \left (a \,x^{k}+b \right )^{3}} &=0 \end {align*}
Solving the above ODE (this ode solved using Maple, not this program), gives
\[ u \left (x \right ) = c_{1} x^{\frac {k \sqrt {\frac {b^{2} n^{2}+2 \beta n b -4 \alpha \gamma +\beta ^{2}}{k^{2} a^{2}}}\, a +b n +\beta }{2 b}} \left (a \,x^{k}+b \right )^{-\frac {k \sqrt {\frac {b^{2} n^{2}+2 \beta n b -4 \alpha \gamma +\beta ^{2}}{k^{2} a^{2}}}\, a +b n +\beta }{2 b k}}+c_{2} x^{\frac {-k \sqrt {\frac {b^{2} n^{2}+2 \beta n b -4 \alpha \gamma +\beta ^{2}}{k^{2} a^{2}}}\, a +b n +\beta }{2 b}} \left (a \,x^{k}+b \right )^{\frac {k \sqrt {\frac {b^{2} n^{2}+2 \beta n b -4 \alpha \gamma +\beta ^{2}}{k^{2} a^{2}}}\, a -b n -\beta }{2 b k}} \] The above shows that \[ u^{\prime }\left (x \right ) = \frac {-x^{\frac {-k \sqrt {\frac {b^{2} n^{2}+2 \beta n b -4 \alpha \gamma +\beta ^{2}}{k^{2} a^{2}}}\, a +b n +\beta }{2 b}} c_{2} \left (k \sqrt {\frac {b^{2} n^{2}+2 \beta n b -4 \alpha \gamma +\beta ^{2}}{k^{2} a^{2}}}\, a -b n -\beta \right ) \left (a \,x^{k}+b \right )^{\frac {k \sqrt {\frac {b^{2} n^{2}+2 \beta n b -4 \alpha \gamma +\beta ^{2}}{k^{2} a^{2}}}\, a -b n -\beta }{2 b k}}+c_{1} x^{\frac {k \sqrt {\frac {b^{2} n^{2}+2 \beta n b -4 \alpha \gamma +\beta ^{2}}{k^{2} a^{2}}}\, a +b n +\beta }{2 b}} \left (k \sqrt {\frac {b^{2} n^{2}+2 \beta n b -4 \alpha \gamma +\beta ^{2}}{k^{2} a^{2}}}\, a +b n +\beta \right ) \left (a \,x^{k}+b \right )^{-\frac {k \sqrt {\frac {b^{2} n^{2}+2 \beta n b -4 \alpha \gamma +\beta ^{2}}{k^{2} a^{2}}}\, a +b n +\beta }{2 b k}}}{2 x \left (a \,x^{k}+b \right )} \] Using the above in (1) gives the solution \[ y = -\frac {\left (-x^{\frac {-k \sqrt {\frac {b^{2} n^{2}+2 \beta n b -4 \alpha \gamma +\beta ^{2}}{k^{2} a^{2}}}\, a +b n +\beta }{2 b}} c_{2} \left (k \sqrt {\frac {b^{2} n^{2}+2 \beta n b -4 \alpha \gamma +\beta ^{2}}{k^{2} a^{2}}}\, a -b n -\beta \right ) \left (a \,x^{k}+b \right )^{\frac {k \sqrt {\frac {b^{2} n^{2}+2 \beta n b -4 \alpha \gamma +\beta ^{2}}{k^{2} a^{2}}}\, a -b n -\beta }{2 b k}}+c_{1} x^{\frac {k \sqrt {\frac {b^{2} n^{2}+2 \beta n b -4 \alpha \gamma +\beta ^{2}}{k^{2} a^{2}}}\, a +b n +\beta }{2 b}} \left (k \sqrt {\frac {b^{2} n^{2}+2 \beta n b -4 \alpha \gamma +\beta ^{2}}{k^{2} a^{2}}}\, a +b n +\beta \right ) \left (a \,x^{k}+b \right )^{-\frac {k \sqrt {\frac {b^{2} n^{2}+2 \beta n b -4 \alpha \gamma +\beta ^{2}}{k^{2} a^{2}}}\, a +b n +\beta }{2 b k}}\right ) x^{-n}}{2 \alpha \left (c_{1} x^{\frac {k \sqrt {\frac {b^{2} n^{2}+2 \beta n b -4 \alpha \gamma +\beta ^{2}}{k^{2} a^{2}}}\, a +b n +\beta }{2 b}} \left (a \,x^{k}+b \right )^{-\frac {k \sqrt {\frac {b^{2} n^{2}+2 \beta n b -4 \alpha \gamma +\beta ^{2}}{k^{2} a^{2}}}\, a +b n +\beta }{2 b k}}+c_{2} x^{\frac {-k \sqrt {\frac {b^{2} n^{2}+2 \beta n b -4 \alpha \gamma +\beta ^{2}}{k^{2} a^{2}}}\, a +b n +\beta }{2 b}} \left (a \,x^{k}+b \right )^{\frac {k \sqrt {\frac {b^{2} n^{2}+2 \beta n b -4 \alpha \gamma +\beta ^{2}}{k^{2} a^{2}}}\, a -b n -\beta }{2 b k}}\right )} \] Dividing both numerator and denominator by \(c_{1}\) gives, after renaming the constant \(\frac {c_{2}}{c_{1}}=c_{3}\) the following solution
\[ y = \frac {\left (a \,x^{k}+b \right )^{\frac {b n +\beta }{2 b k}} x^{-\frac {3 b n +\beta }{2 b}} \left (x^{\frac {-k \sqrt {\frac {b^{2} n^{2}+2 \beta n b -4 \alpha \gamma +\beta ^{2}}{k^{2} a^{2}}}\, a +b n +\beta }{2 b}} \left (k \sqrt {\frac {b^{2} n^{2}+2 \beta n b -4 \alpha \gamma +\beta ^{2}}{k^{2} a^{2}}}\, a -b n -\beta \right ) \left (a \,x^{k}+b \right )^{\frac {k \sqrt {\frac {b^{2} n^{2}+2 \beta n b -4 \alpha \gamma +\beta ^{2}}{k^{2} a^{2}}}\, a -b n -\beta }{2 b k}}-c_{3} x^{\frac {k \sqrt {\frac {b^{2} n^{2}+2 \beta n b -4 \alpha \gamma +\beta ^{2}}{k^{2} a^{2}}}\, a +b n +\beta }{2 b}} \left (k \sqrt {\frac {b^{2} n^{2}+2 \beta n b -4 \alpha \gamma +\beta ^{2}}{k^{2} a^{2}}}\, a +b n +\beta \right ) \left (a \,x^{k}+b \right )^{-\frac {k \sqrt {\frac {b^{2} n^{2}+2 \beta n b -4 \alpha \gamma +\beta ^{2}}{k^{2} a^{2}}}\, a +b n +\beta }{2 b k}}\right )}{2 \alpha \left (\left (a \,x^{k}+b \right )^{-\frac {\sqrt {\frac {b^{2} n^{2}+2 \beta n b -4 \alpha \gamma +\beta ^{2}}{k^{2} a^{2}}}\, a}{2 b}} x^{\frac {k \sqrt {\frac {b^{2} n^{2}+2 \beta n b -4 \alpha \gamma +\beta ^{2}}{k^{2} a^{2}}}\, a}{2 b}} c_{3} +\left (a \,x^{k}+b \right )^{\frac {\sqrt {\frac {b^{2} n^{2}+2 \beta n b -4 \alpha \gamma +\beta ^{2}}{k^{2} a^{2}}}\, a}{2 b}} x^{-\frac {k \sqrt {\frac {b^{2} n^{2}+2 \beta n b -4 \alpha \gamma +\beta ^{2}}{k^{2} a^{2}}}\, a}{2 b}}\right )} \]
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {\left (a \,x^{k}+b \right )^{\frac {b n +\beta }{2 b k}} x^{-\frac {3 b n +\beta }{2 b}} \left (x^{\frac {-k \sqrt {\frac {b^{2} n^{2}+2 \beta n b -4 \alpha \gamma +\beta ^{2}}{k^{2} a^{2}}}\, a +b n +\beta }{2 b}} \left (k \sqrt {\frac {b^{2} n^{2}+2 \beta n b -4 \alpha \gamma +\beta ^{2}}{k^{2} a^{2}}}\, a -b n -\beta \right ) \left (a \,x^{k}+b \right )^{\frac {k \sqrt {\frac {b^{2} n^{2}+2 \beta n b -4 \alpha \gamma +\beta ^{2}}{k^{2} a^{2}}}\, a -b n -\beta }{2 b k}}-c_{3} x^{\frac {k \sqrt {\frac {b^{2} n^{2}+2 \beta n b -4 \alpha \gamma +\beta ^{2}}{k^{2} a^{2}}}\, a +b n +\beta }{2 b}} \left (k \sqrt {\frac {b^{2} n^{2}+2 \beta n b -4 \alpha \gamma +\beta ^{2}}{k^{2} a^{2}}}\, a +b n +\beta \right ) \left (a \,x^{k}+b \right )^{-\frac {k \sqrt {\frac {b^{2} n^{2}+2 \beta n b -4 \alpha \gamma +\beta ^{2}}{k^{2} a^{2}}}\, a +b n +\beta }{2 b k}}\right )}{2 \alpha \left (\left (a \,x^{k}+b \right )^{-\frac {\sqrt {\frac {b^{2} n^{2}+2 \beta n b -4 \alpha \gamma +\beta ^{2}}{k^{2} a^{2}}}\, a}{2 b}} x^{\frac {k \sqrt {\frac {b^{2} n^{2}+2 \beta n b -4 \alpha \gamma +\beta ^{2}}{k^{2} a^{2}}}\, a}{2 b}} c_{3} +\left (a \,x^{k}+b \right )^{\frac {\sqrt {\frac {b^{2} n^{2}+2 \beta n b -4 \alpha \gamma +\beta ^{2}}{k^{2} a^{2}}}\, a}{2 b}} x^{-\frac {k \sqrt {\frac {b^{2} n^{2}+2 \beta n b -4 \alpha \gamma +\beta ^{2}}{k^{2} a^{2}}}\, a}{2 b}}\right )} \\ \end{align*}
Verification of solutions
\[ y = \frac {\left (a \,x^{k}+b \right )^{\frac {b n +\beta }{2 b k}} x^{-\frac {3 b n +\beta }{2 b}} \left (x^{\frac {-k \sqrt {\frac {b^{2} n^{2}+2 \beta n b -4 \alpha \gamma +\beta ^{2}}{k^{2} a^{2}}}\, a +b n +\beta }{2 b}} \left (k \sqrt {\frac {b^{2} n^{2}+2 \beta n b -4 \alpha \gamma +\beta ^{2}}{k^{2} a^{2}}}\, a -b n -\beta \right ) \left (a \,x^{k}+b \right )^{\frac {k \sqrt {\frac {b^{2} n^{2}+2 \beta n b -4 \alpha \gamma +\beta ^{2}}{k^{2} a^{2}}}\, a -b n -\beta }{2 b k}}-c_{3} x^{\frac {k \sqrt {\frac {b^{2} n^{2}+2 \beta n b -4 \alpha \gamma +\beta ^{2}}{k^{2} a^{2}}}\, a +b n +\beta }{2 b}} \left (k \sqrt {\frac {b^{2} n^{2}+2 \beta n b -4 \alpha \gamma +\beta ^{2}}{k^{2} a^{2}}}\, a +b n +\beta \right ) \left (a \,x^{k}+b \right )^{-\frac {k \sqrt {\frac {b^{2} n^{2}+2 \beta n b -4 \alpha \gamma +\beta ^{2}}{k^{2} a^{2}}}\, a +b n +\beta }{2 b k}}\right )}{2 \alpha \left (\left (a \,x^{k}+b \right )^{-\frac {\sqrt {\frac {b^{2} n^{2}+2 \beta n b -4 \alpha \gamma +\beta ^{2}}{k^{2} a^{2}}}\, a}{2 b}} x^{\frac {k \sqrt {\frac {b^{2} n^{2}+2 \beta n b -4 \alpha \gamma +\beta ^{2}}{k^{2} a^{2}}}\, a}{2 b}} c_{3} +\left (a \,x^{k}+b \right )^{\frac {\sqrt {\frac {b^{2} n^{2}+2 \beta n b -4 \alpha \gamma +\beta ^{2}}{k^{2} a^{2}}}\, a}{2 b}} x^{-\frac {k \sqrt {\frac {b^{2} n^{2}+2 \beta n b -4 \alpha \gamma +\beta ^{2}}{k^{2} a^{2}}}\, a}{2 b}}\right )} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x \left (a \,x^{k}+b \right ) y^{\prime }-\alpha \,x^{n} y^{2}-\left (\beta -a n \,x^{k}\right ) y=\gamma \,x^{-n} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {\alpha \,x^{n} y^{2}+\left (\beta -a n \,x^{k}\right ) y+\gamma \,x^{-n}}{x \left (a \,x^{k}+b \right )} \end {array} \]
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 <- Chini successful`
✓ Solution by Maple
Time used: 0.032 (sec). Leaf size: 138
dsolve(x*(a*x^k+b)*diff(y(x),x)=alpha*x^n*y(x)^2+(beta-a*n*x^k)*y(x)+gamma*x^(-n),y(x), singsol=all)
\[ y \left (x \right ) = -\frac {x^{-n} \left (\tanh \left (\frac {\left (\left (-b n -\beta \right ) \ln \left (a \,x^{k}+b \right )+\left (\left (b n +\beta \right ) \ln \left (x \right )+c_{1} b \right ) k \right ) \sqrt {\left (b n +\beta \right )^{2} \left (n^{2} b^{2}+2 b \beta n -4 \gamma \alpha +\beta ^{2}\right )}}{2 k b \left (b n +\beta \right )^{2}}\right ) \sqrt {\left (b n +\beta \right )^{2} \left (n^{2} b^{2}+2 b \beta n -4 \gamma \alpha +\beta ^{2}\right )}+\left (b n +\beta \right )^{2}\right )}{2 \alpha \left (b n +\beta \right )} \]
✓ Solution by Mathematica
Time used: 4.641 (sec). Leaf size: 663
DSolve[x*(a*x^k+b)*y'[x]==\[Alpha]*x^n*y[x]^2+(\[Beta]-a*n*x^k)*y[x]+\[Gamma]*x^(-n),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {x^{-n} \left (b \left (n \left (-\exp \left (-\frac {\left (\log \left (a x^k+b\right )+\log (b)-k \log (x)+\log (k)\right ) \left (\sqrt {\alpha } \sqrt {\gamma } \sqrt {\frac {-4 \alpha \gamma +b^2 n^2+\beta ^2+2 b \beta n}{\alpha \gamma }}+b n+\beta \right )}{2 b k}\right )\right )-c_1 n \exp \left (-\frac {\left (\log \left (a x^k+b\right )+\log (b)-k \log (x)+\log (k)\right ) \left (-\sqrt {\alpha } \sqrt {\gamma } \sqrt {\frac {-4 \alpha \gamma +b^2 n^2+\beta ^2+2 b \beta n}{\alpha \gamma }}+b n+\beta \right )}{2 b k}\right )\right )+\exp \left (-\frac {(b n+\beta ) \left (\log \left (a x^k+b\right )+\log (b)-k \log (x)+\log (k)\right )}{b k}\right ) \left (\left (\sqrt {\alpha } \sqrt {\gamma } \sqrt {\frac {-4 \alpha \gamma +b^2 n^2+\beta ^2+2 b \beta n}{\alpha \gamma }}+\beta \right ) \left (-\exp \left (\frac {\left (\log \left (a x^k+b\right )+\log (b)-k \log (x)+\log (k)\right ) \left (-\sqrt {\alpha } \sqrt {\gamma } \sqrt {\frac {-4 \alpha \gamma +b^2 n^2+\beta ^2+2 b \beta n}{\alpha \gamma }}+b n+\beta \right )}{2 b k}\right )\right )-c_1 \left (\beta -\sqrt {\alpha } \sqrt {\gamma } \sqrt {\frac {-4 \alpha \gamma +b^2 n^2+\beta ^2+2 b \beta n}{\alpha \gamma }}\right ) \exp \left (\frac {\left (\log \left (a x^k+b\right )+\log (b)-k \log (x)+\log (k)\right ) \left (\sqrt {\alpha } \sqrt {\gamma } \sqrt {\frac {-4 \alpha \gamma +b^2 n^2+\beta ^2+2 b \beta n}{\alpha \gamma }}+b n+\beta \right )}{2 b k}\right )\right )\right )}{2 \alpha \left (\exp \left (-\frac {\left (\log \left (a x^k+b\right )+\log (b)-k \log (x)+\log (k)\right ) \left (\sqrt {\alpha } \sqrt {\gamma } \sqrt {\frac {(b n+\beta )^2}{\alpha \gamma }-4}+b n+\beta \right )}{2 b k}\right )+c_1 \exp \left (-\frac {\left (\log \left (a x^k+b\right )+\log (b)-k \log (x)+\log (k)\right ) \left (-\sqrt {\alpha } \sqrt {\gamma } \sqrt {\frac {(b n+\beta )^2}{\alpha \gamma }-4}+b n+\beta \right )}{2 b k}\right )\right )} \\ y(x)\to \frac {x^{-n} \left (\sqrt {\alpha } \sqrt {\gamma } \sqrt {\frac {(b n+\beta )^2}{\alpha \gamma }-4}-b n-\beta \right )}{2 \alpha } \\ \end{align*}