Internal problem ID [10371]
Internal file name [OUTPUT/9319_Monday_June_06_2022_01_51_22_PM_29239198/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: 42.
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]
\[ \boxed {y^{\prime } x -a \,x^{n} y^{2}-y b=c \,x^{m}} \]
In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= \frac {a \,x^{n} y^{2}+y b +c \,x^{m}}{x} \end {align*}
This is a Riccati ODE. Comparing the ODE to solve \[ y' = \frac {a \,x^{n} y^{2}}{x}+\frac {c \,x^{m}}{x}+\frac {y b}{x} \] With Riccati ODE standard form \[ y' = f_0(x)+ f_1(x)y+f_2(x)y^{2} \] Shows that \(f_0(x)=\frac {c \,x^{m}}{x}\), \(f_1(x)=\frac {b}{x}\) and \(f_2(x)=\frac {a \,x^{n}}{x}\). Let \begin {align*} y &= \frac {-u'}{f_2 u} \\ &= \frac {-u'}{\frac {a \,x^{n} u}{x}} \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 \,x^{n}}{x^{2}}+\frac {a \,x^{n} n}{x^{2}}\\ f_1 f_2 &=\frac {b a \,x^{n}}{x^{2}}\\ f_2^2 f_0 &=\frac {a^{2} x^{2 n} c \,x^{m}}{x^{3}} \end {align*}
Substituting the above terms back in equation (2) gives \begin {align*} \frac {a \,x^{n} u^{\prime \prime }\left (x \right )}{x}-\left (-\frac {a \,x^{n}}{x^{2}}+\frac {a \,x^{n} n}{x^{2}}+\frac {b a \,x^{n}}{x^{2}}\right ) u^{\prime }\left (x \right )+\frac {a^{2} x^{2 n} c \,x^{m} u \left (x \right )}{x^{3}} &=0 \end {align*}
Solving the above ODE (this ode solved using Maple, not this program), gives
\[ u \left (x \right ) = \left (\operatorname {BesselY}\left (\frac {-b -n}{m +n}, \frac {2 \sqrt {c a}\, x^{\frac {m}{2}+\frac {n}{2}}}{m +n}\right ) c_{2} +\operatorname {BesselJ}\left (\frac {-b -n}{m +n}, \frac {2 \sqrt {c a}\, x^{\frac {m}{2}+\frac {n}{2}}}{m +n}\right ) c_{1} \right ) x^{\frac {b}{2}+\frac {n}{2}} \] The above shows that \[ u^{\prime }\left (x \right ) = x^{-1+\frac {b}{2}+n +\frac {m}{2}} \sqrt {c a}\, \left (-\operatorname {BesselY}\left (\frac {-b +m}{m +n}, \frac {2 \sqrt {c a}\, x^{\frac {m}{2}+\frac {n}{2}}}{m +n}\right ) c_{2} -\operatorname {BesselJ}\left (\frac {-b +m}{m +n}, \frac {2 \sqrt {c a}\, x^{\frac {m}{2}+\frac {n}{2}}}{m +n}\right ) c_{1} \right ) \] Using the above in (1) gives the solution \[ y = -\frac {x^{-1+\frac {b}{2}+n +\frac {m}{2}} \sqrt {c a}\, \left (-\operatorname {BesselY}\left (\frac {-b +m}{m +n}, \frac {2 \sqrt {c a}\, x^{\frac {m}{2}+\frac {n}{2}}}{m +n}\right ) c_{2} -\operatorname {BesselJ}\left (\frac {-b +m}{m +n}, \frac {2 \sqrt {c a}\, x^{\frac {m}{2}+\frac {n}{2}}}{m +n}\right ) c_{1} \right ) x \,x^{-n} x^{-\frac {b}{2}-\frac {n}{2}}}{a \left (\operatorname {BesselY}\left (\frac {-b -n}{m +n}, \frac {2 \sqrt {c a}\, x^{\frac {m}{2}+\frac {n}{2}}}{m +n}\right ) c_{2} +\operatorname {BesselJ}\left (\frac {-b -n}{m +n}, \frac {2 \sqrt {c a}\, x^{\frac {m}{2}+\frac {n}{2}}}{m +n}\right ) c_{1} \right )} \] Dividing both numerator and denominator by \(c_{1}\) gives, after renaming the constant \(\frac {c_{2}}{c_{1}}=c_{3}\) the following solution
\[ y = \frac {x^{\frac {m}{2}-\frac {n}{2}} \sqrt {c a}\, \left (\operatorname {BesselJ}\left (\frac {-b +m}{m +n}, \frac {2 \sqrt {c a}\, x^{\frac {m}{2}+\frac {n}{2}}}{m +n}\right ) c_{3} +\operatorname {BesselY}\left (\frac {-b +m}{m +n}, \frac {2 \sqrt {c a}\, x^{\frac {m}{2}+\frac {n}{2}}}{m +n}\right )\right )}{a \left (\operatorname {BesselY}\left (\frac {-b -n}{m +n}, \frac {2 \sqrt {c a}\, x^{\frac {m}{2}+\frac {n}{2}}}{m +n}\right )+\operatorname {BesselJ}\left (\frac {-b -n}{m +n}, \frac {2 \sqrt {c a}\, x^{\frac {m}{2}+\frac {n}{2}}}{m +n}\right ) c_{3} \right )} \]
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {x^{\frac {m}{2}-\frac {n}{2}} \sqrt {c a}\, \left (\operatorname {BesselJ}\left (\frac {-b +m}{m +n}, \frac {2 \sqrt {c a}\, x^{\frac {m}{2}+\frac {n}{2}}}{m +n}\right ) c_{3} +\operatorname {BesselY}\left (\frac {-b +m}{m +n}, \frac {2 \sqrt {c a}\, x^{\frac {m}{2}+\frac {n}{2}}}{m +n}\right )\right )}{a \left (\operatorname {BesselY}\left (\frac {-b -n}{m +n}, \frac {2 \sqrt {c a}\, x^{\frac {m}{2}+\frac {n}{2}}}{m +n}\right )+\operatorname {BesselJ}\left (\frac {-b -n}{m +n}, \frac {2 \sqrt {c a}\, x^{\frac {m}{2}+\frac {n}{2}}}{m +n}\right ) c_{3} \right )} \\ \end{align*}
Verification of solutions
\[ y = \frac {x^{\frac {m}{2}-\frac {n}{2}} \sqrt {c a}\, \left (\operatorname {BesselJ}\left (\frac {-b +m}{m +n}, \frac {2 \sqrt {c a}\, x^{\frac {m}{2}+\frac {n}{2}}}{m +n}\right ) c_{3} +\operatorname {BesselY}\left (\frac {-b +m}{m +n}, \frac {2 \sqrt {c a}\, x^{\frac {m}{2}+\frac {n}{2}}}{m +n}\right )\right )}{a \left (\operatorname {BesselY}\left (\frac {-b -n}{m +n}, \frac {2 \sqrt {c a}\, x^{\frac {m}{2}+\frac {n}{2}}}{m +n}\right )+\operatorname {BesselJ}\left (\frac {-b -n}{m +n}, \frac {2 \sqrt {c a}\, x^{\frac {m}{2}+\frac {n}{2}}}{m +n}\right ) c_{3} \right )} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime } x -a \,x^{n} y^{2}-y b =c \,x^{m} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {a \,x^{n} y^{2}+y b +c \,x^{m}}{x} \end {array} \]
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+n-1)*(diff(y(x), x))/x-x^(n-1)*a*x^(m-1)*c*y(x), y(x)` *** Sub 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 -> Trying an equivalence, under non-integer power transformations, to LODEs admitting Liouvillian solutions. -> Trying a Liouvillian solution using Kovacics algorithm <- No Liouvillian solutions exists -> Trying a solution in terms of special functions: -> Bessel <- Bessel successful <- special function solution successful <- Riccati to 2nd Order successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 166
dsolve(x*diff(y(x),x)=a*x^(n)*y(x)^2+b*y(x)+c*x^(m),y(x), singsol=all)
\[ y \left (x \right ) = \frac {x^{-\frac {n}{2}+\frac {m}{2}} \sqrt {a c}\, \left (\operatorname {BesselY}\left (\frac {-b +m}{n +m}, \frac {2 \sqrt {a c}\, x^{\frac {m}{2}+\frac {n}{2}}}{n +m}\right ) c_{1} +\operatorname {BesselJ}\left (\frac {-b +m}{n +m}, \frac {2 \sqrt {a c}\, x^{\frac {m}{2}+\frac {n}{2}}}{n +m}\right )\right )}{a \left (\operatorname {BesselY}\left (\frac {-b -n}{n +m}, \frac {2 \sqrt {a c}\, x^{\frac {m}{2}+\frac {n}{2}}}{n +m}\right ) c_{1} +\operatorname {BesselJ}\left (\frac {-b -n}{n +m}, \frac {2 \sqrt {a c}\, x^{\frac {m}{2}+\frac {n}{2}}}{n +m}\right )\right )} \]
✓ Solution by Mathematica
Time used: 1.49 (sec). Leaf size: 1321
DSolve[x*y'[x]==a*x^(n)*y[x]^2+b*y[x]+c*x^(m),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {x^{-n} \left (\sqrt {a} \sqrt {c} (m+n) x^{m+n} \operatorname {BesselJ}\left (\frac {m-b}{m+n},\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^{m+n}}}{\sqrt {(m+n)^2}}\right ) c_1 \operatorname {Gamma}\left (\frac {m-b}{m+n}\right ) \left ((m+n)^2\right )^{\frac {b+n}{m+n}}-\sqrt {a} \sqrt {c} m x^{m+n} \operatorname {BesselJ}\left (-\frac {b+m+2 n}{m+n},\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^{m+n}}}{\sqrt {(m+n)^2}}\right ) c_1 \operatorname {Gamma}\left (\frac {m-b}{m+n}\right ) \left ((m+n)^2\right )^{\frac {b+n}{m+n}}-\sqrt {a} \sqrt {c} n x^{m+n} \operatorname {BesselJ}\left (-\frac {b+m+2 n}{m+n},\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^{m+n}}}{\sqrt {(m+n)^2}}\right ) c_1 \operatorname {Gamma}\left (\frac {m-b}{m+n}\right ) \left ((m+n)^2\right )^{\frac {b+n}{m+n}}-(b+n) \sqrt {x^{m+n}} \operatorname {BesselJ}\left (-\frac {b+n}{m+n},\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^{m+n}}}{\sqrt {(m+n)^2}}\right ) c_1 \operatorname {Gamma}\left (\frac {m-b}{m+n}\right ) \left ((m+n)^2\right )^{\frac {2 b+m+3 n}{2 (m+n)}}-b (m+n)^{\frac {2 (b+n)}{m+n}} \sqrt {x^{m+n}} \operatorname {BesselJ}\left (\frac {b+n}{m+n},\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^{m+n}}}{\sqrt {(m+n)^2}}\right ) \operatorname {Gamma}\left (\frac {b+m+2 n}{m+n}\right ) \sqrt {(m+n)^2}-n (m+n)^{\frac {2 (b+n)}{m+n}} \sqrt {x^{m+n}} \operatorname {BesselJ}\left (\frac {b+n}{m+n},\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^{m+n}}}{\sqrt {(m+n)^2}}\right ) \operatorname {Gamma}\left (\frac {b+m+2 n}{m+n}\right ) \sqrt {(m+n)^2}-\sqrt {a} \sqrt {c} m (m+n)^{\frac {2 (b+n)}{m+n}} x^{m+n} \operatorname {BesselJ}\left (\frac {b-m}{m+n},\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^{m+n}}}{\sqrt {(m+n)^2}}\right ) \operatorname {Gamma}\left (\frac {b+m+2 n}{m+n}\right )-\sqrt {a} \sqrt {c} n (m+n)^{\frac {2 (b+n)}{m+n}} x^{m+n} \operatorname {BesselJ}\left (\frac {b-m}{m+n},\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^{m+n}}}{\sqrt {(m+n)^2}}\right ) \operatorname {Gamma}\left (\frac {b+m+2 n}{m+n}\right )+\sqrt {a} \sqrt {c} m (m+n)^{\frac {2 (b+n)}{m+n}} x^{m+n} \operatorname {BesselJ}\left (\frac {b+n}{m+n}+1,\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^{m+n}}}{\sqrt {(m+n)^2}}\right ) \operatorname {Gamma}\left (\frac {b+m+2 n}{m+n}\right )+\sqrt {a} \sqrt {c} n (m+n)^{\frac {2 (b+n)}{m+n}} x^{m+n} \operatorname {BesselJ}\left (\frac {b+n}{m+n}+1,\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^{m+n}}}{\sqrt {(m+n)^2}}\right ) \operatorname {Gamma}\left (\frac {b+m+2 n}{m+n}\right )\right )}{2 a \sqrt {(m+n)^2} \sqrt {x^{m+n}} \left (\operatorname {BesselJ}\left (-\frac {b+n}{m+n},\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^{m+n}}}{\sqrt {(m+n)^2}}\right ) c_1 \operatorname {Gamma}\left (\frac {m-b}{m+n}\right ) \left ((m+n)^2\right )^{\frac {b+n}{m+n}}+(m+n)^{\frac {2 (b+n)}{m+n}} \operatorname {BesselJ}\left (\frac {b+n}{m+n},\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^{m+n}}}{\sqrt {(m+n)^2}}\right ) \operatorname {Gamma}\left (\frac {b+m+2 n}{m+n}\right )\right )} \\ y(x)\to \frac {x^{-n} \left (\sqrt {a} \sqrt {c} (m+n) \sqrt {x^{m+n}} \operatorname {BesselJ}\left (\frac {m-b}{m+n},\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^{m+n}}}{\sqrt {(m+n)^2}}\right )-(b+n) \sqrt {(m+n)^2} \operatorname {BesselJ}\left (-\frac {b+n}{m+n},\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^{m+n}}}{\sqrt {(m+n)^2}}\right )-\sqrt {a} \sqrt {c} (m+n) \sqrt {x^{m+n}} \operatorname {BesselJ}\left (-\frac {b+m+2 n}{m+n},\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^{m+n}}}{\sqrt {(m+n)^2}}\right )\right )}{2 a \sqrt {(m+n)^2} \operatorname {BesselJ}\left (-\frac {b+n}{m+n},\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^{m+n}}}{\sqrt {(m+n)^2}}\right )} \\ \end{align*}