Internal problem ID [10385]
Internal file name [OUTPUT/9333_Monday_June_06_2022_01_56_50_PM_34885414/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: 56.
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]
\[ \boxed {\left (x^{2} a +b \right ) y^{\prime }+\alpha y^{2}+\beta x y=-\frac {b \left (a +\beta \right )}{\alpha }} \]
In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= -\frac {\alpha ^{2} y^{2}+\beta x y \alpha +a b +\beta b}{\left (x^{2} a +b \right ) \alpha } \end {align*}
This is a Riccati ODE. Comparing the ODE to solve \[ y' = -\frac {\alpha \,y^{2}}{x^{2} a +b}-\frac {\beta x y}{x^{2} a +b}-\frac {b a}{\left (x^{2} a +b \right ) \alpha }-\frac {b \beta }{\left (x^{2} a +b \right ) \alpha } \] With Riccati ODE standard form \[ y' = f_0(x)+ f_1(x)y+f_2(x)y^{2} \] Shows that \(f_0(x)=-\frac {a b +\beta b}{\left (x^{2} a +b \right ) \alpha }\), \(f_1(x)=-\frac {\beta x}{x^{2} a +b}\) and \(f_2(x)=-\frac {\alpha }{x^{2} a +b}\). Let \begin {align*} y &= \frac {-u'}{f_2 u} \\ &= \frac {-u'}{-\frac {\alpha u}{x^{2} a +b}} \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 \alpha x a}{\left (x^{2} a +b \right )^{2}}\\ f_1 f_2 &=\frac {\beta x \alpha }{\left (x^{2} a +b \right )^{2}}\\ f_2^2 f_0 &=-\frac {\alpha \left (a b +\beta b \right )}{\left (x^{2} a +b \right )^{3}} \end {align*}
Substituting the above terms back in equation (2) gives \begin {align*} -\frac {\alpha u^{\prime \prime }\left (x \right )}{x^{2} a +b}-\left (\frac {\beta x \alpha }{\left (x^{2} a +b \right )^{2}}+\frac {2 \alpha x a}{\left (x^{2} a +b \right )^{2}}\right ) u^{\prime }\left (x \right )-\frac {\alpha \left (a b +\beta b \right ) u \left (x \right )}{\left (x^{2} a +b \right )^{3}} &=0 \end {align*}
Solving the above ODE (this ode solved using Maple, not this program), gives
\[ u \left (x \right ) = \left (x^{2} a +b \right )^{-\frac {\beta }{4 a}} \left (\operatorname {LegendreP}\left (\frac {\beta }{2 a}, \frac {2 a +\beta }{2 a}, \frac {a x}{\sqrt {-a b}}\right ) c_{1} +\operatorname {LegendreQ}\left (\frac {\beta }{2 a}, \frac {2 a +\beta }{2 a}, \frac {a x}{\sqrt {-a b}}\right ) c_{2} \right ) \] The above shows that \[ u^{\prime }\left (x \right ) = x \left (a +\beta \right ) \left (x^{2} a +b \right )^{-\frac {4 a +\beta }{4 a}} \left (-\operatorname {LegendreP}\left (\frac {\beta }{2 a}, \frac {2 a +\beta }{2 a}, \frac {a x}{\sqrt {-a b}}\right ) c_{1} -\operatorname {LegendreQ}\left (\frac {\beta }{2 a}, \frac {2 a +\beta }{2 a}, \frac {a x}{\sqrt {-a b}}\right ) c_{2} \right ) \] Using the above in (1) gives the solution \[ y = \frac {x \left (a +\beta \right ) \left (x^{2} a +b \right )^{-\frac {4 a +\beta }{4 a}} \left (-\operatorname {LegendreP}\left (\frac {\beta }{2 a}, \frac {2 a +\beta }{2 a}, \frac {a x}{\sqrt {-a b}}\right ) c_{1} -\operatorname {LegendreQ}\left (\frac {\beta }{2 a}, \frac {2 a +\beta }{2 a}, \frac {a x}{\sqrt {-a b}}\right ) c_{2} \right ) \left (x^{2} a +b \right ) \left (x^{2} a +b \right )^{\frac {\beta }{4 a}}}{\alpha \left (\operatorname {LegendreP}\left (\frac {\beta }{2 a}, \frac {2 a +\beta }{2 a}, \frac {a x}{\sqrt {-a b}}\right ) c_{1} +\operatorname {LegendreQ}\left (\frac {\beta }{2 a}, \frac {2 a +\beta }{2 a}, \frac {a x}{\sqrt {-a b}}\right ) c_{2} \right )} \] Dividing both numerator and denominator by \(c_{1}\) gives, after renaming the constant \(\frac {c_{2}}{c_{1}}=c_{3}\) the following solution
\[ y = -\frac {\left (a +\beta \right ) x}{\alpha } \]
The solution(s) found are the following \begin{align*} \tag{1} y &= -\frac {\left (a +\beta \right ) x}{\alpha } \\ \end{align*}
Verification of solutions
\[ y = -\frac {\left (a +\beta \right ) x}{\alpha } \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime } a \alpha \,x^{2}+\alpha ^{2} y^{2}+\beta x y \alpha +y^{\prime } \alpha b +a b +\beta b =0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-\frac {\alpha ^{2} y^{2}+\beta x y \alpha +a b +\beta b}{\alpha \,x^{2} a +\alpha b} \end {array} \]
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: <- Abel AIR successful: ODE belongs to the 2F1 3-parameter class`
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 517
dsolve((a*x^2+b)*diff(y(x),x)+alpha*y(x)^2+beta*x*y(x)+b/alpha*(a+beta)=0,y(x), singsol=all)
\[ y \left (x \right ) = -\frac {b \,a^{2} \left (-\frac {\left (-\frac {-a x +\sqrt {-a b}}{2 \sqrt {-a b}}\right )^{\frac {\beta }{a}} \left (a \,x^{2}+b \right ) \left (a \,x^{2}+2 \sqrt {-a b}\, x -b \right ) \operatorname {HeunCPrime}\left (0, -1-\frac {\beta }{a}, 1+\frac {\beta }{2 a}, 0, \frac {1}{2}+\frac {\beta }{2 a}+\frac {\beta ^{2}}{4 a^{2}}, \frac {2 \sqrt {-a b}}{-a x +\sqrt {-a b}}\right )}{2}-2 c_{1} b \left (\left (3 a \,x^{2}-b \right ) \sqrt {-a b}+x a \left (a \,x^{2}-3 b \right )\right ) a \operatorname {HeunCPrime}\left (0, \frac {\beta }{a}+1, 1+\frac {\beta }{2 a}, 0, \frac {1}{2}+\frac {\beta }{2 a}+\frac {\beta ^{2}}{4 a^{2}}, \frac {2 \sqrt {-a b}}{-a x +\sqrt {-a b}}\right )+\left (a \,x^{2}+b \right ) \left (\frac {\left (-\frac {-a x +\sqrt {-a b}}{2 \sqrt {-a b}}\right )^{\frac {\beta }{a}} \left (a \,x^{2}-2 \sqrt {-a b}\, x -b \right ) \operatorname {hypergeom}\left (\left [1, -\frac {\beta }{2 a}\right ], \left [-\frac {\beta }{a}\right ], \frac {2 \sqrt {-a b}}{a x +\sqrt {-a b}}\right )}{4}+c_{1} \left (\frac {a x -\sqrt {-a b}}{a x +\sqrt {-a b}}\right )^{\frac {\beta }{2 a}} \left (\left (-a^{2} x^{2}+\left (-x^{2} \beta -2 b \right ) a -b \beta \right ) \sqrt {-a b}+a^{2} b x \right )\right )\right )}{\left (-\frac {\left (-\frac {-a x +\sqrt {-a b}}{2 \sqrt {-a b}}\right )^{\frac {\beta }{a}} \sqrt {-a b}\, \left (a \,x^{2}+b \right ) \operatorname {hypergeom}\left (\left [1, -\frac {\beta }{2 a}\right ], \left [-\frac {\beta }{a}\right ], \frac {2 \sqrt {-a b}}{a x +\sqrt {-a b}}\right )}{4}+\left (\frac {a x -\sqrt {-a b}}{a x +\sqrt {-a b}}\right )^{\frac {\beta }{2 a}} a^{2} b c_{1} \left (-\sqrt {-a b}\, x +b \right )\right ) \left (a x -\sqrt {-a b}\right )^{2} \alpha } \]
✓ Solution by Mathematica
Time used: 1.111 (sec). Leaf size: 27
DSolve[(a*x^2+b)*y'[x]+\[Alpha]*y[x]^2+\[Beta]*x*y[x]+b/\[Alpha]*(a+\[Beta])==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {x (a+\beta )}{\alpha } \\ y(x)\to -\frac {x (a+\beta )}{\alpha } \\ \end{align*}