Internal problem ID [8367]
Internal file name [OUTPUT/7300_Sunday_June_05_2022_05_43_38_PM_18025179/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 30.
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]
\[ \boxed {y^{\prime }+x^{-a -1} y^{2}=x^{a}} \]
In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= -x^{-a -1} y^{2}+x^{a} \end {align*}
This is a Riccati ODE. Comparing the ODE to solve \[ y' = -\frac {x^{-a} y^{2}}{x}+x^{a} \] With Riccati ODE standard form \[ y' = f_0(x)+ f_1(x)y+f_2(x)y^{2} \] Shows that \(f_0(x)=x^{a}\), \(f_1(x)=0\) and \(f_2(x)=-x^{-a -1}\). Let \begin {align*} y &= \frac {-u'}{f_2 u} \\ &= \frac {-u'}{-x^{-a -1} 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' &=-\frac {x^{-a -1} \left (-a -1\right )}{x}\\ f_1 f_2 &=0\\ f_2^2 f_0 &=x^{-2 a -2} x^{a} \end {align*}
Substituting the above terms back in equation (2) gives \begin {align*} -x^{-a -1} u^{\prime \prime }\left (x \right )+\frac {x^{-a -1} \left (-a -1\right ) u^{\prime }\left (x \right )}{x}+x^{-2 a -2} x^{a} u \left (x \right ) &=0 \end {align*}
Solving the above ODE (this ode solved using Maple, not this program), gives
\[ u \left (x \right ) = \left (\operatorname {BesselI}\left (a , 2 \sqrt {x}\right ) c_{1} +\operatorname {BesselK}\left (a , 2 \sqrt {x}\right ) c_{2} \right ) x^{-\frac {a}{2}} \] The above shows that \[ u^{\prime }\left (x \right ) = x^{-\frac {a}{2}-\frac {1}{2}} \left (\operatorname {BesselI}\left (a +1, 2 \sqrt {x}\right ) c_{1} -\operatorname {BesselK}\left (a +1, 2 \sqrt {x}\right ) c_{2} \right ) \] Using the above in (1) gives the solution \[ y = \frac {x^{-\frac {a}{2}-\frac {1}{2}} \left (\operatorname {BesselI}\left (a +1, 2 \sqrt {x}\right ) c_{1} -\operatorname {BesselK}\left (a +1, 2 \sqrt {x}\right ) c_{2} \right ) x^{a +1} x^{\frac {a}{2}}}{\operatorname {BesselI}\left (a , 2 \sqrt {x}\right ) c_{1} +\operatorname {BesselK}\left (a , 2 \sqrt {x}\right ) c_{2}} \] Dividing both numerator and denominator by \(c_{1}\) gives, after renaming the constant \(\frac {c_{2}}{c_{1}}=c_{3}\) the following solution
\[ y = \frac {x^{\frac {1}{2}+a} \left (\operatorname {BesselI}\left (a +1, 2 \sqrt {x}\right ) c_{3} -\operatorname {BesselK}\left (a +1, 2 \sqrt {x}\right )\right )}{\operatorname {BesselI}\left (a , 2 \sqrt {x}\right ) c_{3} +\operatorname {BesselK}\left (a , 2 \sqrt {x}\right )} \]
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {x^{\frac {1}{2}+a} \left (\operatorname {BesselI}\left (a +1, 2 \sqrt {x}\right ) c_{3} -\operatorname {BesselK}\left (a +1, 2 \sqrt {x}\right )\right )}{\operatorname {BesselI}\left (a , 2 \sqrt {x}\right ) c_{3} +\operatorname {BesselK}\left (a , 2 \sqrt {x}\right )} \\ \end{align*}
Verification of solutions
\[ y = \frac {x^{\frac {1}{2}+a} \left (\operatorname {BesselI}\left (a +1, 2 \sqrt {x}\right ) c_{3} -\operatorname {BesselK}\left (a +1, 2 \sqrt {x}\right )\right )}{\operatorname {BesselI}\left (a , 2 \sqrt {x}\right ) c_{3} +\operatorname {BesselK}\left (a , 2 \sqrt {x}\right )} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }+x^{-a -1} y^{2}=x^{a} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-x^{-a -1} y^{2}+x^{a} \end {array} \]
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) = -(a+1)*(diff(y(x), x))/x+x^(-a-1)*x^a*y(x), y(x)` *** Sublevel 2 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 <- Bessel successful <- special function solution successful <- Riccati to 2nd Order successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 51
dsolve(diff(y(x),x) + x^(-a-1)*y(x)^2 - x^a=0,y(x), singsol=all)
\[ y \left (x \right ) = \frac {x^{\frac {1}{2}+a} \left (-\operatorname {BesselK}\left (a +1, 2 \sqrt {x}\right ) c_{1} +\operatorname {BesselI}\left (a +1, 2 \sqrt {x}\right )\right )}{\operatorname {BesselK}\left (a , 2 \sqrt {x}\right ) c_{1} +\operatorname {BesselI}\left (a , 2 \sqrt {x}\right )} \]
✓ Solution by Mathematica
Time used: 0.363 (sec). Leaf size: 265
DSolve[y'[x] + x^(-a-1)*y[x]^2 - x^a==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {x^a \left (\sqrt {x} \operatorname {Gamma}(1-a) \operatorname {BesselI}\left (-a-1,2 \sqrt {x}\right )+\sqrt {x} \operatorname {Gamma}(1-a) \operatorname {BesselI}\left (1-a,2 \sqrt {x}\right )-a \operatorname {Gamma}(1-a) \operatorname {BesselI}\left (-a,2 \sqrt {x}\right )+(-1)^a c_1 \sqrt {x} \operatorname {Gamma}(a+1) \operatorname {BesselI}\left (a-1,2 \sqrt {x}\right )-(-1)^a a c_1 \operatorname {Gamma}(a+1) \operatorname {BesselI}\left (a,2 \sqrt {x}\right )+(-1)^a c_1 \sqrt {x} \operatorname {Gamma}(a+1) \operatorname {BesselI}\left (a+1,2 \sqrt {x}\right )\right )}{2 \left (\operatorname {Gamma}(1-a) \operatorname {BesselI}\left (-a,2 \sqrt {x}\right )+(-1)^a c_1 \operatorname {Gamma}(a+1) \operatorname {BesselI}\left (a,2 \sqrt {x}\right )\right )} \\ y(x)\to \frac {x^a \left (\sqrt {x} \operatorname {BesselI}\left (a-1,2 \sqrt {x}\right )-a \operatorname {BesselI}\left (a,2 \sqrt {x}\right )+\sqrt {x} \operatorname {BesselI}\left (a+1,2 \sqrt {x}\right )\right )}{2 \operatorname {BesselI}\left (a,2 \sqrt {x}\right )} \\ \end{align*}