Internal problem ID [10395]
Internal file name [OUTPUT/9343_Monday_June_06_2022_02_14_08_PM_49204472/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: 66.
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]
\[ \boxed {x^{3} y^{\prime }-a \,x^{3} y^{2}-x \left (b x +c \right ) y=\alpha x +\beta } \]
In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= \frac {a \,x^{3} y^{2}+b \,x^{2} y +y x c +\alpha x +\beta }{x^{3}} \end {align*}
This is a Riccati ODE. Comparing the ODE to solve \[ y' = a \,y^{2}+\frac {y b}{x}+\frac {y c}{x^{2}}+\frac {\alpha }{x^{2}}+\frac {\beta }{x^{3}} \] With Riccati ODE standard form \[ y' = f_0(x)+ f_1(x)y+f_2(x)y^{2} \] Shows that \(f_0(x)=\frac {\alpha x +\beta }{x^{3}}\), \(f_1(x)=\frac {b \,x^{2}+c x}{x^{3}}\) and \(f_2(x)=a\). Let \begin {align*} y &= \frac {-u'}{f_2 u} \\ &= \frac {-u'}{a 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 &=\frac {\left (b \,x^{2}+c x \right ) a}{x^{3}}\\ f_2^2 f_0 &=\frac {a^{2} \left (\alpha x +\beta \right )}{x^{3}} \end {align*}
Substituting the above terms back in equation (2) gives \begin {align*} a u^{\prime \prime }\left (x \right )-\frac {\left (b \,x^{2}+c x \right ) a u^{\prime }\left (x \right )}{x^{3}}+\frac {a^{2} \left (\alpha x +\beta \right ) u \left (x \right )}{x^{3}} &=0 \end {align*}
Solving the above ODE (this ode solved using Maple, not this program), gives
\[ u \left (x \right ) = x^{\frac {b}{2}+\frac {1}{2}-\frac {\sqrt {-4 \alpha a +b^{2}+2 b +1}}{2}} {\mathrm e}^{-\frac {c}{x}} \left (\operatorname {KummerM}\left (\frac {\sqrt {-4 \alpha a +b^{2}+2 b +1}\, c +\left (b +3\right ) c -2 \beta a}{2 c}, 1+\sqrt {-4 \alpha a +b^{2}+2 b +1}, \frac {c}{x}\right ) c_{1} +\operatorname {KummerU}\left (\frac {\sqrt {-4 \alpha a +b^{2}+2 b +1}\, c +\left (b +3\right ) c -2 \beta a}{2 c}, 1+\sqrt {-4 \alpha a +b^{2}+2 b +1}, \frac {c}{x}\right ) c_{2} \right ) \] The above shows that \[ u^{\prime }\left (x \right ) = -\frac {\left (c_{2} x \left (\left (\alpha a +b +2\right ) c^{2}-a \beta \left (b +3\right ) c +a^{2} \beta ^{2}\right ) \operatorname {KummerU}\left (\frac {\sqrt {-4 \alpha a +b^{2}+2 b +1}\, c +\left (b +5\right ) c -2 \beta a}{2 c}, 1+\sqrt {-4 \alpha a +b^{2}+2 b +1}, \frac {c}{x}\right )+c \left (c_{1} \left (-c^{2}-x \left (2+b \right ) c +\beta a x \right ) \operatorname {KummerM}\left (\frac {\sqrt {-4 \alpha a +b^{2}+2 b +1}\, c +\left (b +3\right ) c -2 \beta a}{2 c}, 1+\sqrt {-4 \alpha a +b^{2}+2 b +1}, \frac {c}{x}\right )-\left (-\frac {\sqrt {-4 \alpha a +b^{2}+2 b +1}\, c}{2}+\frac {\left (-b -3\right ) c}{2}+\beta a \right ) c_{1} x \operatorname {KummerM}\left (\frac {\sqrt {-4 \alpha a +b^{2}+2 b +1}\, c +\left (b +5\right ) c -2 \beta a}{2 c}, 1+\sqrt {-4 \alpha a +b^{2}+2 b +1}, \frac {c}{x}\right )+\operatorname {KummerU}\left (\frac {\sqrt {-4 \alpha a +b^{2}+2 b +1}\, c +\left (b +3\right ) c -2 \beta a}{2 c}, 1+\sqrt {-4 \alpha a +b^{2}+2 b +1}, \frac {c}{x}\right ) c_{2} \left (-c^{2}-x \left (2+b \right ) c +\beta a x \right )\right )\right ) x^{-\frac {3}{2}+\frac {b}{2}-\frac {\sqrt {-4 \alpha a +b^{2}+2 b +1}}{2}} {\mathrm e}^{-\frac {c}{x}}}{c^{2}} \] Using the above in (1) gives the solution \[ y = \frac {\left (c_{2} x \left (\left (\alpha a +b +2\right ) c^{2}-a \beta \left (b +3\right ) c +a^{2} \beta ^{2}\right ) \operatorname {KummerU}\left (\frac {\sqrt {-4 \alpha a +b^{2}+2 b +1}\, c +\left (b +5\right ) c -2 \beta a}{2 c}, 1+\sqrt {-4 \alpha a +b^{2}+2 b +1}, \frac {c}{x}\right )+c \left (c_{1} \left (-c^{2}-x \left (2+b \right ) c +\beta a x \right ) \operatorname {KummerM}\left (\frac {\sqrt {-4 \alpha a +b^{2}+2 b +1}\, c +\left (b +3\right ) c -2 \beta a}{2 c}, 1+\sqrt {-4 \alpha a +b^{2}+2 b +1}, \frac {c}{x}\right )-\left (-\frac {\sqrt {-4 \alpha a +b^{2}+2 b +1}\, c}{2}+\frac {\left (-b -3\right ) c}{2}+\beta a \right ) c_{1} x \operatorname {KummerM}\left (\frac {\sqrt {-4 \alpha a +b^{2}+2 b +1}\, c +\left (b +5\right ) c -2 \beta a}{2 c}, 1+\sqrt {-4 \alpha a +b^{2}+2 b +1}, \frac {c}{x}\right )+\operatorname {KummerU}\left (\frac {\sqrt {-4 \alpha a +b^{2}+2 b +1}\, c +\left (b +3\right ) c -2 \beta a}{2 c}, 1+\sqrt {-4 \alpha a +b^{2}+2 b +1}, \frac {c}{x}\right ) c_{2} \left (-c^{2}-x \left (2+b \right ) c +\beta a x \right )\right )\right ) x^{-\frac {3}{2}+\frac {b}{2}-\frac {\sqrt {-4 \alpha a +b^{2}+2 b +1}}{2}} x^{\frac {\sqrt {-4 \alpha a +b^{2}+2 b +1}}{2}-\frac {b}{2}-\frac {1}{2}}}{c^{2} a \left (\operatorname {KummerM}\left (\frac {\sqrt {-4 \alpha a +b^{2}+2 b +1}\, c +\left (b +3\right ) c -2 \beta a}{2 c}, 1+\sqrt {-4 \alpha a +b^{2}+2 b +1}, \frac {c}{x}\right ) c_{1} +\operatorname {KummerU}\left (\frac {\sqrt {-4 \alpha a +b^{2}+2 b +1}\, c +\left (b +3\right ) c -2 \beta a}{2 c}, 1+\sqrt {-4 \alpha a +b^{2}+2 b +1}, \frac {c}{x}\right ) c_{2} \right )} \] Dividing both numerator and denominator by \(c_{1}\) gives, after renaming the constant \(\frac {c_{2}}{c_{1}}=c_{3}\) the following solution
\[ y = \frac {\left (\left (\alpha a +b +2\right ) c^{2}-a \beta \left (b +3\right ) c +a^{2} \beta ^{2}\right ) x \operatorname {KummerU}\left (\frac {\sqrt {-4 \alpha a +b^{2}+2 b +1}\, c +\left (b +5\right ) c -2 \beta a}{2 c}, 1+\sqrt {-4 \alpha a +b^{2}+2 b +1}, \frac {c}{x}\right )-c \left (-c_{3} \left (-c^{2}-x \left (2+b \right ) c +\beta a x \right ) \operatorname {KummerM}\left (\frac {\sqrt {-4 \alpha a +b^{2}+2 b +1}\, c +\left (b +3\right ) c -2 \beta a}{2 c}, 1+\sqrt {-4 \alpha a +b^{2}+2 b +1}, \frac {c}{x}\right )+\left (c^{2}+x \left (2+b \right ) c -\beta a x \right ) \operatorname {KummerU}\left (\frac {\sqrt {-4 \alpha a +b^{2}+2 b +1}\, c +\left (b +3\right ) c -2 \beta a}{2 c}, 1+\sqrt {-4 \alpha a +b^{2}+2 b +1}, \frac {c}{x}\right )+\left (-\frac {\sqrt {-4 \alpha a +b^{2}+2 b +1}\, c}{2}+\frac {\left (-b -3\right ) c}{2}+\beta a \right ) c_{3} x \operatorname {KummerM}\left (\frac {\sqrt {-4 \alpha a +b^{2}+2 b +1}\, c +\left (b +5\right ) c -2 \beta a}{2 c}, 1+\sqrt {-4 \alpha a +b^{2}+2 b +1}, \frac {c}{x}\right )\right )}{a \,c^{2} x^{2} \left (\operatorname {KummerM}\left (\frac {\sqrt {-4 \alpha a +b^{2}+2 b +1}\, c +\left (b +3\right ) c -2 \beta a}{2 c}, 1+\sqrt {-4 \alpha a +b^{2}+2 b +1}, \frac {c}{x}\right ) c_{3} +\operatorname {KummerU}\left (\frac {\sqrt {-4 \alpha a +b^{2}+2 b +1}\, c +\left (b +3\right ) c -2 \beta a}{2 c}, 1+\sqrt {-4 \alpha a +b^{2}+2 b +1}, \frac {c}{x}\right )\right )} \]
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {\left (\left (\alpha a +b +2\right ) c^{2}-a \beta \left (b +3\right ) c +a^{2} \beta ^{2}\right ) x \operatorname {KummerU}\left (\frac {\sqrt {-4 \alpha a +b^{2}+2 b +1}\, c +\left (b +5\right ) c -2 \beta a}{2 c}, 1+\sqrt {-4 \alpha a +b^{2}+2 b +1}, \frac {c}{x}\right )-c \left (-c_{3} \left (-c^{2}-x \left (2+b \right ) c +\beta a x \right ) \operatorname {KummerM}\left (\frac {\sqrt {-4 \alpha a +b^{2}+2 b +1}\, c +\left (b +3\right ) c -2 \beta a}{2 c}, 1+\sqrt {-4 \alpha a +b^{2}+2 b +1}, \frac {c}{x}\right )+\left (c^{2}+x \left (2+b \right ) c -\beta a x \right ) \operatorname {KummerU}\left (\frac {\sqrt {-4 \alpha a +b^{2}+2 b +1}\, c +\left (b +3\right ) c -2 \beta a}{2 c}, 1+\sqrt {-4 \alpha a +b^{2}+2 b +1}, \frac {c}{x}\right )+\left (-\frac {\sqrt {-4 \alpha a +b^{2}+2 b +1}\, c}{2}+\frac {\left (-b -3\right ) c}{2}+\beta a \right ) c_{3} x \operatorname {KummerM}\left (\frac {\sqrt {-4 \alpha a +b^{2}+2 b +1}\, c +\left (b +5\right ) c -2 \beta a}{2 c}, 1+\sqrt {-4 \alpha a +b^{2}+2 b +1}, \frac {c}{x}\right )\right )}{a \,c^{2} x^{2} \left (\operatorname {KummerM}\left (\frac {\sqrt {-4 \alpha a +b^{2}+2 b +1}\, c +\left (b +3\right ) c -2 \beta a}{2 c}, 1+\sqrt {-4 \alpha a +b^{2}+2 b +1}, \frac {c}{x}\right ) c_{3} +\operatorname {KummerU}\left (\frac {\sqrt {-4 \alpha a +b^{2}+2 b +1}\, c +\left (b +3\right ) c -2 \beta a}{2 c}, 1+\sqrt {-4 \alpha a +b^{2}+2 b +1}, \frac {c}{x}\right )\right )} \\ \end{align*}
Verification of solutions
\[ y = \frac {\left (\left (\alpha a +b +2\right ) c^{2}-a \beta \left (b +3\right ) c +a^{2} \beta ^{2}\right ) x \operatorname {KummerU}\left (\frac {\sqrt {-4 \alpha a +b^{2}+2 b +1}\, c +\left (b +5\right ) c -2 \beta a}{2 c}, 1+\sqrt {-4 \alpha a +b^{2}+2 b +1}, \frac {c}{x}\right )-c \left (-c_{3} \left (-c^{2}-x \left (2+b \right ) c +\beta a x \right ) \operatorname {KummerM}\left (\frac {\sqrt {-4 \alpha a +b^{2}+2 b +1}\, c +\left (b +3\right ) c -2 \beta a}{2 c}, 1+\sqrt {-4 \alpha a +b^{2}+2 b +1}, \frac {c}{x}\right )+\left (c^{2}+x \left (2+b \right ) c -\beta a x \right ) \operatorname {KummerU}\left (\frac {\sqrt {-4 \alpha a +b^{2}+2 b +1}\, c +\left (b +3\right ) c -2 \beta a}{2 c}, 1+\sqrt {-4 \alpha a +b^{2}+2 b +1}, \frac {c}{x}\right )+\left (-\frac {\sqrt {-4 \alpha a +b^{2}+2 b +1}\, c}{2}+\frac {\left (-b -3\right ) c}{2}+\beta a \right ) c_{3} x \operatorname {KummerM}\left (\frac {\sqrt {-4 \alpha a +b^{2}+2 b +1}\, c +\left (b +5\right ) c -2 \beta a}{2 c}, 1+\sqrt {-4 \alpha a +b^{2}+2 b +1}, \frac {c}{x}\right )\right )}{a \,c^{2} x^{2} \left (\operatorname {KummerM}\left (\frac {\sqrt {-4 \alpha a +b^{2}+2 b +1}\, c +\left (b +3\right ) c -2 \beta a}{2 c}, 1+\sqrt {-4 \alpha a +b^{2}+2 b +1}, \frac {c}{x}\right ) c_{3} +\operatorname {KummerU}\left (\frac {\sqrt {-4 \alpha a +b^{2}+2 b +1}\, c +\left (b +3\right ) c -2 \beta a}{2 c}, 1+\sqrt {-4 \alpha a +b^{2}+2 b +1}, \frac {c}{x}\right )\right )} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x^{3} y^{\prime }-a \,x^{3} y^{2}-x \left (b x +c \right ) y=\alpha x +\beta \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {a \,x^{3} y^{2}+x \left (b x +c \right ) y+\alpha x +\beta }{x^{3}} \end {array} \]
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 to 2nd Order -> Calling odsolve with the ODE`, diff(diff(y(x), x), x) = (b*x+c)*(diff(y(x), x))/x^2-a*(alpha*x+beta)*y(x)/x^3, y(x)` *** Methods for second order ODEs: --- Trying classification methods --- trying a quadrature checking if the LODE has constant coefficients checking if the LODE is of Euler type trying a symmetry of the form [xi=0, eta=F(x)] checking if the LODE is missing y -> Trying a Liouvillian solution using Kovacics algorithm <- No Liouvillian solutions exists -> Trying a solution in terms of special functions: -> Bessel -> elliptic -> Legendre -> Kummer -> hyper3: Equivalence to 1F1 under a power @ Moebius <- hyper3 successful: received ODE is equivalent to the 1F1 ODE <- Kummer successful <- special function solution successful <- Riccati to 2nd Order successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 438
dsolve(x^3*diff(y(x),x)=a*x^3*y(x)^2+x*(b*x+c)*y(x)+alpha*x+beta,y(x), singsol=all)
\[ y \left (x \right ) = \frac {\left (\left (a \alpha +b +2\right ) c^{2}-a \beta \left (b +3\right ) c +a^{2} \beta ^{2}\right ) x c_{1} \operatorname {KummerU}\left (\frac {\sqrt {-4 a \alpha +b^{2}+2 b +1}\, c +\left (b +5\right ) c -2 \beta a}{2 c}, 1+\sqrt {-4 a \alpha +b^{2}+2 b +1}, \frac {c}{x}\right )-c \left (\left (c^{2}+x \left (b +2\right ) c -a x \beta \right ) \operatorname {KummerM}\left (\frac {\sqrt {-4 a \alpha +b^{2}+2 b +1}\, c +c \left (b +3\right )-2 \beta a}{2 c}, 1+\sqrt {-4 a \alpha +b^{2}+2 b +1}, \frac {c}{x}\right )-c_{1} \left (-c^{2}-x \left (b +2\right ) c +a x \beta \right ) \operatorname {KummerU}\left (\frac {\sqrt {-4 a \alpha +b^{2}+2 b +1}\, c +c \left (b +3\right )-2 \beta a}{2 c}, 1+\sqrt {-4 a \alpha +b^{2}+2 b +1}, \frac {c}{x}\right )+x \operatorname {KummerM}\left (\frac {\sqrt {-4 a \alpha +b^{2}+2 b +1}\, c +\left (b +5\right ) c -2 \beta a}{2 c}, 1+\sqrt {-4 a \alpha +b^{2}+2 b +1}, \frac {c}{x}\right ) \left (-\frac {\sqrt {-4 a \alpha +b^{2}+2 b +1}\, c}{2}+\frac {\left (-b -3\right ) c}{2}+\beta a \right )\right )}{c^{2} x^{2} a \left (\operatorname {KummerU}\left (\frac {\sqrt {-4 a \alpha +b^{2}+2 b +1}\, c +c \left (b +3\right )-2 \beta a}{2 c}, 1+\sqrt {-4 a \alpha +b^{2}+2 b +1}, \frac {c}{x}\right ) c_{1} +\operatorname {KummerM}\left (\frac {\sqrt {-4 a \alpha +b^{2}+2 b +1}\, c +c \left (b +3\right )-2 \beta a}{2 c}, 1+\sqrt {-4 a \alpha +b^{2}+2 b +1}, \frac {c}{x}\right )\right )} \]
✓ Solution by Mathematica
Time used: 2.395 (sec). Leaf size: 908
DSolve[x^3*y'[x]==a*x^3*y[x]^2+x*(b*x+c)*y[x]+\[Alpha]*x+\[Beta],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {\frac {c^{\sqrt {-4 a \alpha +b^2+2 b+1}} \left (\frac {1}{x}\right )^{\sqrt {-4 a \alpha +b^2+2 b+1}+1} \left (c \left (\sqrt {-4 a \alpha +b^2+2 b+1}-b-1\right )+2 a \beta \right ) \operatorname {Hypergeometric1F1}\left (\frac {1}{2} \left (-b+\frac {2 a \beta }{c}+\sqrt {b^2+2 b-4 a \alpha +1}+1\right ),\sqrt {b^2+2 b-4 a \alpha +1}+2,-\frac {c}{x}\right )}{\sqrt {-4 a \alpha +b^2+2 b+1}+1}-\left (\sqrt {-4 a \alpha +b^2+2 b+1}-b-1\right ) c^{\sqrt {-4 a \alpha +b^2+2 b+1}} \left (\frac {1}{x}\right )^{\sqrt {-4 a \alpha +b^2+2 b+1}} \operatorname {Hypergeometric1F1}\left (\frac {1}{2} \left (-b+\frac {2 a \beta }{c}+\sqrt {b^2+2 b-4 a \alpha +1}-1\right ),\sqrt {b^2+2 b-4 a \alpha +1}+1,-\frac {c}{x}\right )+\frac {c_1 \left (\left (c \left (\sqrt {-4 a \alpha +b^2+2 b+1}+b+1\right )-2 a \beta \right ) \operatorname {Hypergeometric1F1}\left (\frac {2 a \beta -c \left (b+\sqrt {b^2+2 b-4 a \alpha +1}-1\right )}{2 c},2-\sqrt {b^2+2 b-4 a \alpha +1},-\frac {c}{x}\right )+x \left (-4 a \alpha +b \sqrt {-4 a \alpha +b^2+2 b+1}+b^2+b\right ) \operatorname {Hypergeometric1F1}\left (\frac {2 a \beta -c \left (b+\sqrt {b^2+2 b-4 a \alpha +1}+1\right )}{2 c},1-\sqrt {b^2+2 b-4 a \alpha +1},-\frac {c}{x}\right )\right )}{x \left (\sqrt {-4 a \alpha +b^2+2 b+1}-1\right )}}{2 a x \left (c^{\sqrt {-4 a \alpha +b^2+2 b+1}} \left (\frac {1}{x}\right )^{\sqrt {-4 a \alpha +b^2+2 b+1}} \operatorname {Hypergeometric1F1}\left (\frac {1}{2} \left (-b+\frac {2 a \beta }{c}+\sqrt {b^2+2 b-4 a \alpha +1}-1\right ),\sqrt {b^2+2 b-4 a \alpha +1}+1,-\frac {c}{x}\right )+c_1 \operatorname {Hypergeometric1F1}\left (\frac {2 a \beta -c \left (b+\sqrt {b^2+2 b-4 a \alpha +1}+1\right )}{2 c},1-\sqrt {b^2+2 b-4 a \alpha +1},-\frac {c}{x}\right )\right )} \\ y(x)\to \frac {\frac {\left (c \left (b \left (\sqrt {-4 a \alpha +b^2+2 b+1}+3\right )+2 \left (-2 a \alpha +\sqrt {-4 a \alpha +b^2+2 b+1}+1\right )+b^2\right )-2 a \beta \left (\sqrt {-4 a \alpha +b^2+2 b+1}+1\right )\right ) \operatorname {Hypergeometric1F1}\left (\frac {2 a \beta -c \left (b+\sqrt {b^2+2 b-4 a \alpha +1}-1\right )}{2 c},2-\sqrt {b^2+2 b-4 a \alpha +1},-\frac {c}{x}\right )}{\operatorname {Hypergeometric1F1}\left (\frac {2 a \beta -c \left (b+\sqrt {b^2+2 b-4 a \alpha +1}+1\right )}{2 c},1-\sqrt {b^2+2 b-4 a \alpha +1},-\frac {c}{x}\right )}+x \left (-4 a \alpha +b^2+2 b\right ) \left (\sqrt {-4 a \alpha +b^2+2 b+1}+b+1\right )}{2 a x^2 \left (4 a \alpha -b^2-2 b\right )} \\ \end{align*}