Internal problem ID [10495]
Internal file name [OUTPUT/9443_Monday_June_06_2022_02_32_37_PM_81719794/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. subsection 1.2.5-2
Problem number: 21.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "riccati"
Maple gives the following as the ode type
[_Riccati]
\[ \boxed {x^{2} y^{\prime }-y^{2} a^{2} x^{2}+x y=b^{2} \ln \left (x \right )^{n}} \]
In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= \frac {y^{2} a^{2} x^{2}-y x +b^{2} \ln \left (x \right )^{n}}{x^{2}} \end {align*}
This is a Riccati ODE. Comparing the ODE to solve \[ y' = a^{2} y^{2}+\frac {b^{2} \ln \left (x \right )^{n}}{x^{2}}-\frac {y}{x} \] With Riccati ODE standard form \[ y' = f_0(x)+ f_1(x)y+f_2(x)y^{2} \] Shows that \(f_0(x)=\frac {b^{2} \ln \left (x \right )^{n}}{x^{2}}\), \(f_1(x)=-\frac {1}{x}\) and \(f_2(x)=a^{2}\). Let \begin {align*} y &= \frac {-u'}{f_2 u} \\ &= \frac {-u'}{a^{2} 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 {a^{2}}{x}\\ f_2^2 f_0 &=\frac {a^{4} b^{2} \ln \left (x \right )^{n}}{x^{2}} \end {align*}
Substituting the above terms back in equation (2) gives \begin {align*} a^{2} u^{\prime \prime }\left (x \right )+\frac {a^{2} u^{\prime }\left (x \right )}{x}+\frac {a^{4} b^{2} \ln \left (x \right )^{n} u \left (x \right )}{x^{2}} &=0 \end {align*}
Solving the above ODE (this ode solved using Maple, not this program), gives
\[ u \left (x \right ) = \left (\operatorname {BesselY}\left (\frac {1}{2+n}, \frac {2 \sqrt {b^{2} a^{2}}\, \ln \left (x \right )^{1+\frac {n}{2}}}{2+n}\right ) c_{2} +\operatorname {BesselJ}\left (\frac {1}{2+n}, \frac {2 \sqrt {b^{2} a^{2}}\, \ln \left (x \right )^{1+\frac {n}{2}}}{2+n}\right ) c_{1} \right ) \sqrt {\ln \left (x \right )} \] The above shows that \[ u^{\prime }\left (x \right ) = \frac {-\sqrt {b^{2} a^{2}}\, \operatorname {BesselJ}\left (\frac {n +3}{2+n}, \frac {2 \sqrt {b^{2} a^{2}}\, \ln \left (x \right )^{1+\frac {n}{2}}}{2+n}\right ) \ln \left (x \right )^{1+\frac {n}{2}} c_{1} -\sqrt {b^{2} a^{2}}\, \operatorname {BesselY}\left (\frac {n +3}{2+n}, \frac {2 \sqrt {b^{2} a^{2}}\, \ln \left (x \right )^{1+\frac {n}{2}}}{2+n}\right ) \ln \left (x \right )^{1+\frac {n}{2}} c_{2} +\operatorname {BesselY}\left (\frac {1}{2+n}, \frac {2 \sqrt {b^{2} a^{2}}\, \ln \left (x \right )^{1+\frac {n}{2}}}{2+n}\right ) c_{2} +\operatorname {BesselJ}\left (\frac {1}{2+n}, \frac {2 \sqrt {b^{2} a^{2}}\, \ln \left (x \right )^{1+\frac {n}{2}}}{2+n}\right ) c_{1}}{\sqrt {\ln \left (x \right )}\, x} \] Using the above in (1) gives the solution \[ y = -\frac {-\sqrt {b^{2} a^{2}}\, \operatorname {BesselJ}\left (\frac {n +3}{2+n}, \frac {2 \sqrt {b^{2} a^{2}}\, \ln \left (x \right )^{1+\frac {n}{2}}}{2+n}\right ) \ln \left (x \right )^{1+\frac {n}{2}} c_{1} -\sqrt {b^{2} a^{2}}\, \operatorname {BesselY}\left (\frac {n +3}{2+n}, \frac {2 \sqrt {b^{2} a^{2}}\, \ln \left (x \right )^{1+\frac {n}{2}}}{2+n}\right ) \ln \left (x \right )^{1+\frac {n}{2}} c_{2} +\operatorname {BesselY}\left (\frac {1}{2+n}, \frac {2 \sqrt {b^{2} a^{2}}\, \ln \left (x \right )^{1+\frac {n}{2}}}{2+n}\right ) c_{2} +\operatorname {BesselJ}\left (\frac {1}{2+n}, \frac {2 \sqrt {b^{2} a^{2}}\, \ln \left (x \right )^{1+\frac {n}{2}}}{2+n}\right ) c_{1}}{\ln \left (x \right ) x \,a^{2} \left (\operatorname {BesselY}\left (\frac {1}{2+n}, \frac {2 \sqrt {b^{2} a^{2}}\, \ln \left (x \right )^{1+\frac {n}{2}}}{2+n}\right ) c_{2} +\operatorname {BesselJ}\left (\frac {1}{2+n}, \frac {2 \sqrt {b^{2} a^{2}}\, \ln \left (x \right )^{1+\frac {n}{2}}}{2+n}\right ) c_{1} \right )} \] Dividing both numerator and denominator by \(c_{1}\) gives, after renaming the constant \(\frac {c_{2}}{c_{1}}=c_{3}\) the following solution
\[ y = \frac {\sqrt {b^{2} a^{2}}\, \operatorname {BesselJ}\left (\frac {n +3}{2+n}, \frac {2 \sqrt {b^{2} a^{2}}\, \ln \left (x \right )^{1+\frac {n}{2}}}{2+n}\right ) \ln \left (x \right )^{1+\frac {n}{2}} c_{3} +\sqrt {b^{2} a^{2}}\, \ln \left (x \right )^{1+\frac {n}{2}} \operatorname {BesselY}\left (\frac {n +3}{2+n}, \frac {2 \sqrt {b^{2} a^{2}}\, \ln \left (x \right )^{1+\frac {n}{2}}}{2+n}\right )-\operatorname {BesselY}\left (\frac {1}{2+n}, \frac {2 \sqrt {b^{2} a^{2}}\, \ln \left (x \right )^{1+\frac {n}{2}}}{2+n}\right )-\operatorname {BesselJ}\left (\frac {1}{2+n}, \frac {2 \sqrt {b^{2} a^{2}}\, \ln \left (x \right )^{1+\frac {n}{2}}}{2+n}\right ) c_{3}}{\ln \left (x \right ) x \,a^{2} \left (\operatorname {BesselY}\left (\frac {1}{2+n}, \frac {2 \sqrt {b^{2} a^{2}}\, \ln \left (x \right )^{1+\frac {n}{2}}}{2+n}\right )+\operatorname {BesselJ}\left (\frac {1}{2+n}, \frac {2 \sqrt {b^{2} a^{2}}\, \ln \left (x \right )^{1+\frac {n}{2}}}{2+n}\right ) c_{3} \right )} \]
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {\sqrt {b^{2} a^{2}}\, \operatorname {BesselJ}\left (\frac {n +3}{2+n}, \frac {2 \sqrt {b^{2} a^{2}}\, \ln \left (x \right )^{1+\frac {n}{2}}}{2+n}\right ) \ln \left (x \right )^{1+\frac {n}{2}} c_{3} +\sqrt {b^{2} a^{2}}\, \ln \left (x \right )^{1+\frac {n}{2}} \operatorname {BesselY}\left (\frac {n +3}{2+n}, \frac {2 \sqrt {b^{2} a^{2}}\, \ln \left (x \right )^{1+\frac {n}{2}}}{2+n}\right )-\operatorname {BesselY}\left (\frac {1}{2+n}, \frac {2 \sqrt {b^{2} a^{2}}\, \ln \left (x \right )^{1+\frac {n}{2}}}{2+n}\right )-\operatorname {BesselJ}\left (\frac {1}{2+n}, \frac {2 \sqrt {b^{2} a^{2}}\, \ln \left (x \right )^{1+\frac {n}{2}}}{2+n}\right ) c_{3}}{\ln \left (x \right ) x \,a^{2} \left (\operatorname {BesselY}\left (\frac {1}{2+n}, \frac {2 \sqrt {b^{2} a^{2}}\, \ln \left (x \right )^{1+\frac {n}{2}}}{2+n}\right )+\operatorname {BesselJ}\left (\frac {1}{2+n}, \frac {2 \sqrt {b^{2} a^{2}}\, \ln \left (x \right )^{1+\frac {n}{2}}}{2+n}\right ) c_{3} \right )} \\ \end{align*}
Verification of solutions
\[ y = \frac {\sqrt {b^{2} a^{2}}\, \operatorname {BesselJ}\left (\frac {n +3}{2+n}, \frac {2 \sqrt {b^{2} a^{2}}\, \ln \left (x \right )^{1+\frac {n}{2}}}{2+n}\right ) \ln \left (x \right )^{1+\frac {n}{2}} c_{3} +\sqrt {b^{2} a^{2}}\, \ln \left (x \right )^{1+\frac {n}{2}} \operatorname {BesselY}\left (\frac {n +3}{2+n}, \frac {2 \sqrt {b^{2} a^{2}}\, \ln \left (x \right )^{1+\frac {n}{2}}}{2+n}\right )-\operatorname {BesselY}\left (\frac {1}{2+n}, \frac {2 \sqrt {b^{2} a^{2}}\, \ln \left (x \right )^{1+\frac {n}{2}}}{2+n}\right )-\operatorname {BesselJ}\left (\frac {1}{2+n}, \frac {2 \sqrt {b^{2} a^{2}}\, \ln \left (x \right )^{1+\frac {n}{2}}}{2+n}\right ) c_{3}}{\ln \left (x \right ) x \,a^{2} \left (\operatorname {BesselY}\left (\frac {1}{2+n}, \frac {2 \sqrt {b^{2} a^{2}}\, \ln \left (x \right )^{1+\frac {n}{2}}}{2+n}\right )+\operatorname {BesselJ}\left (\frac {1}{2+n}, \frac {2 \sqrt {b^{2} a^{2}}\, \ln \left (x \right )^{1+\frac {n}{2}}}{2+n}\right ) c_{3} \right )} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x^{2} y^{\prime }-y^{2} a^{2} x^{2}+x y=b^{2} \ln \left (x \right )^{n} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {y^{2} a^{2} x^{2}-x y+b^{2} \ln \left (x \right )^{n}}{x^{2}} \end {array} \]
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) = -(diff(y(x), x))/x-a^2*b^2*ln(x)^n*y(x)/x^2, y(x)` *** Sublevel 2 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 -> Heun: Equivalence to the GHE or one of its 4 confluent cases under a power @ Moebius -> trying a solution of the form r0(x) * Y + r1(x) * Y where Y = exp(int(r(x), dx)) * 2F1([a1, a2], [b1], f) -> Trying changes of variables to rationalize or make the ODE simpler 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 Change of variables used: [x = exp(t)] Linear ODE actually solved: a^2*b^2*t^n*u(t)+diff(diff(u(t),t),t) = 0 <- change of variables successful <- Riccati to 2nd Order successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 251
dsolve(x^2*diff(y(x),x)=a^2*x^2*y(x)^2-x*y(x)+b^2*(ln(x))^n,y(x), singsol=all)
\[ y \left (x \right ) = \frac {\ln \left (x \right )^{\frac {n}{2}+1} \operatorname {BesselY}\left (\frac {3+n}{n +2}, \frac {2 \sqrt {a^{2} b^{2}}\, \ln \left (x \right )^{\frac {n}{2}+1}}{n +2}\right ) \sqrt {a^{2} b^{2}}\, c_{1} +\operatorname {BesselJ}\left (\frac {3+n}{n +2}, \frac {2 \sqrt {a^{2} b^{2}}\, \ln \left (x \right )^{\frac {n}{2}+1}}{n +2}\right ) \sqrt {a^{2} b^{2}}\, \ln \left (x \right )^{\frac {n}{2}+1}-\operatorname {BesselY}\left (\frac {1}{n +2}, \frac {2 \sqrt {a^{2} b^{2}}\, \ln \left (x \right )^{\frac {n}{2}+1}}{n +2}\right ) c_{1} -\operatorname {BesselJ}\left (\frac {1}{n +2}, \frac {2 \sqrt {a^{2} b^{2}}\, \ln \left (x \right )^{\frac {n}{2}+1}}{n +2}\right )}{\left (\operatorname {BesselY}\left (\frac {1}{n +2}, \frac {2 \sqrt {a^{2} b^{2}}\, \ln \left (x \right )^{\frac {n}{2}+1}}{n +2}\right ) c_{1} +\operatorname {BesselJ}\left (\frac {1}{n +2}, \frac {2 \sqrt {a^{2} b^{2}}\, \ln \left (x \right )^{\frac {n}{2}+1}}{n +2}\right )\right ) a^{2} x \ln \left (x \right )} \]
✓ Solution by Mathematica
Time used: 45.846 (sec). Leaf size: 1769
DSolve[x^2*y'[x]==a^2*x^2*y[x]^2-x*y[x]+b^2*(Log[x])^n,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {2 b^2 \sqrt {b^{\frac {2}{n+1}} (n+2)^2} \left ((n+2)^{\frac {2 (n+1)}{n+2}} \operatorname {BesselJ}\left (\frac {n+1}{n+2},\frac {2 a \sqrt {\left (b^2 \log ^{n+1}(x)\right )^{1+\frac {1}{n+1}}}}{\sqrt {b^{\frac {2}{n+1}} (n+2)^2}}\right ) \operatorname {Gamma}\left (\frac {2 n+3}{n+2}\right ) b^{\frac {2}{n+2}}+\left (b^{\frac {2}{n+1}} (n+2)^2\right )^{\frac {n+1}{n+2}} \operatorname {BesselJ}\left (-\frac {n+1}{n+2},\frac {2 a \sqrt {\left (b^2 \log ^{n+1}(x)\right )^{1+\frac {1}{n+1}}}}{\sqrt {b^{\frac {2}{n+1}} (n+2)^2}}\right ) c_1 \operatorname {Gamma}\left (\frac {1}{n+2}\right )\right ) \log ^{n+1}(x) \sqrt {\left (b^2 \log ^{n+1}(x)\right )^{1+\frac {1}{n+1}}}}{x \left (2 a (n+2)^{\frac {2 (n+1)}{n+2}} \operatorname {BesselJ}\left (-\frac {1}{n+2},\frac {2 a \sqrt {\left (b^2 \log ^{n+1}(x)\right )^{1+\frac {1}{n+1}}}}{\sqrt {b^{\frac {2}{n+1}} (n+2)^2}}\right ) \operatorname {Gamma}\left (\frac {2 n+3}{n+2}\right ) \left (b^2 \log ^{n+1}(x)\right )^{1+\frac {1}{n+1}} b^{\frac {2}{n+2}}+a n (n+2)^{\frac {2 (n+1)}{n+2}} \operatorname {BesselJ}\left (-\frac {1}{n+2},\frac {2 a \sqrt {\left (b^2 \log ^{n+1}(x)\right )^{1+\frac {1}{n+1}}}}{\sqrt {b^{\frac {2}{n+1}} (n+2)^2}}\right ) \operatorname {Gamma}\left (\frac {2 n+3}{n+2}\right ) \left (b^2 \log ^{n+1}(x)\right )^{1+\frac {1}{n+1}} b^{\frac {2}{n+2}}-2 a (n+2)^{\frac {2 (n+1)}{n+2}} \operatorname {BesselJ}\left (\frac {2 n+3}{n+2},\frac {2 a \sqrt {\left (b^2 \log ^{n+1}(x)\right )^{1+\frac {1}{n+1}}}}{\sqrt {b^{\frac {2}{n+1}} (n+2)^2}}\right ) \operatorname {Gamma}\left (\frac {2 n+3}{n+2}\right ) \left (b^2 \log ^{n+1}(x)\right )^{1+\frac {1}{n+1}} b^{\frac {2}{n+2}}-a n (n+2)^{\frac {2 (n+1)}{n+2}} \operatorname {BesselJ}\left (\frac {2 n+3}{n+2},\frac {2 a \sqrt {\left (b^2 \log ^{n+1}(x)\right )^{1+\frac {1}{n+1}}}}{\sqrt {b^{\frac {2}{n+1}} (n+2)^2}}\right ) \operatorname {Gamma}\left (\frac {2 n+3}{n+2}\right ) \left (b^2 \log ^{n+1}(x)\right )^{1+\frac {1}{n+1}} b^{\frac {2}{n+2}}+n (n+2)^{\frac {2 (n+1)}{n+2}} \sqrt {b^{\frac {2}{n+1}} (n+2)^2} \operatorname {BesselJ}\left (\frac {n+1}{n+2},\frac {2 a \sqrt {\left (b^2 \log ^{n+1}(x)\right )^{1+\frac {1}{n+1}}}}{\sqrt {b^{\frac {2}{n+1}} (n+2)^2}}\right ) \operatorname {Gamma}\left (\frac {2 n+3}{n+2}\right ) \sqrt {\left (b^2 \log ^{n+1}(x)\right )^{1+\frac {1}{n+1}}} b^{\frac {2}{n+2}}+(n+2)^{\frac {2 (n+1)}{n+2}} \sqrt {b^{\frac {2}{n+1}} (n+2)^2} \operatorname {BesselJ}\left (\frac {n+1}{n+2},\frac {2 a \sqrt {\left (b^2 \log ^{n+1}(x)\right )^{1+\frac {1}{n+1}}}}{\sqrt {b^{\frac {2}{n+1}} (n+2)^2}}\right ) \operatorname {Gamma}\left (\frac {2 n+3}{n+2}\right ) \sqrt {\left (b^2 \log ^{n+1}(x)\right )^{1+\frac {1}{n+1}}} b^{\frac {2}{n+2}}-a (n+2) \left (b^{\frac {2}{n+1}} (n+2)^2\right )^{\frac {n+1}{n+2}} \operatorname {BesselJ}\left (\frac {1}{n+2},\frac {2 a \sqrt {\left (b^2 \log ^{n+1}(x)\right )^{1+\frac {1}{n+1}}}}{\sqrt {b^{\frac {2}{n+1}} (n+2)^2}}\right ) c_1 \operatorname {Gamma}\left (\frac {1}{n+2}\right ) \left (b^2 \log ^{n+1}(x)\right )^{1+\frac {1}{n+1}}+a (n+2) \left (b^{\frac {2}{n+1}} (n+2)^2\right )^{\frac {n+1}{n+2}} \operatorname {BesselJ}\left (-\frac {2 n+3}{n+2},\frac {2 a \sqrt {\left (b^2 \log ^{n+1}(x)\right )^{1+\frac {1}{n+1}}}}{\sqrt {b^{\frac {2}{n+1}} (n+2)^2}}\right ) c_1 \operatorname {Gamma}\left (\frac {1}{n+2}\right ) \left (b^2 \log ^{n+1}(x)\right )^{1+\frac {1}{n+1}}+n \left (b^{\frac {2}{n+1}} (n+2)^2\right )^{\frac {3 n+4}{2 n+4}} \operatorname {BesselJ}\left (-\frac {n+1}{n+2},\frac {2 a \sqrt {\left (b^2 \log ^{n+1}(x)\right )^{1+\frac {1}{n+1}}}}{\sqrt {b^{\frac {2}{n+1}} (n+2)^2}}\right ) c_1 \operatorname {Gamma}\left (\frac {1}{n+2}\right ) \sqrt {\left (b^2 \log ^{n+1}(x)\right )^{1+\frac {1}{n+1}}}+\left (b^{\frac {2}{n+1}} (n+2)^2\right )^{\frac {3 n+4}{2 n+4}} \operatorname {BesselJ}\left (-\frac {n+1}{n+2},\frac {2 a \sqrt {\left (b^2 \log ^{n+1}(x)\right )^{1+\frac {1}{n+1}}}}{\sqrt {b^{\frac {2}{n+1}} (n+2)^2}}\right ) c_1 \operatorname {Gamma}\left (\frac {1}{n+2}\right ) \sqrt {\left (b^2 \log ^{n+1}(x)\right )^{1+\frac {1}{n+1}}}\right )} \\ y(x)\to \frac {2 b^2 \sqrt {(n+2)^2 b^{\frac {2}{n+1}}} \log ^{n+1}(x) \sqrt {\left (b^2 \log ^{n+1}(x)\right )^{\frac {1}{n+1}+1}} \operatorname {BesselJ}\left (-\frac {n+1}{n+2},\frac {2 a \sqrt {\left (b^2 \log ^{n+1}(x)\right )^{1+\frac {1}{n+1}}}}{\sqrt {b^{\frac {2}{n+1}} (n+2)^2}}\right )}{x \left (-a (n+2) \left (b^2 \log ^{n+1}(x)\right )^{\frac {1}{n+1}+1} \operatorname {BesselJ}\left (\frac {1}{n+2},\frac {2 a \sqrt {\left (b^2 \log ^{n+1}(x)\right )^{1+\frac {1}{n+1}}}}{\sqrt {b^{\frac {2}{n+1}} (n+2)^2}}\right )+a (n+2) \left (b^2 \log ^{n+1}(x)\right )^{\frac {1}{n+1}+1} \operatorname {BesselJ}\left (-\frac {2 n+3}{n+2},\frac {2 a \sqrt {\left (b^2 \log ^{n+1}(x)\right )^{1+\frac {1}{n+1}}}}{\sqrt {b^{\frac {2}{n+1}} (n+2)^2}}\right )+(n+1) \sqrt {(n+2)^2 b^{\frac {2}{n+1}}} \sqrt {\left (b^2 \log ^{n+1}(x)\right )^{\frac {1}{n+1}+1}} \operatorname {BesselJ}\left (-\frac {n+1}{n+2},\frac {2 a \sqrt {\left (b^2 \log ^{n+1}(x)\right )^{1+\frac {1}{n+1}}}}{\sqrt {b^{\frac {2}{n+1}} (n+2)^2}}\right )\right )} \\ \end{align*}