Internal problem ID [8480]
Internal file name [OUTPUT/7413_Sunday_June_05_2022_10_54_42_PM_39711997/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 144.
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]
\[ \boxed {x^{2} \left (y^{\prime }+a y^{2}\right )=-b \,x^{\alpha }-c} \]
In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= -\frac {a \,x^{2} y^{2}+b \,x^{\alpha }+c}{x^{2}} \end {align*}
This is a Riccati ODE. Comparing the ODE to solve \[ y' = -a \,y^{2}-\frac {b \,x^{\alpha }}{x^{2}}-\frac {c}{x^{2}} \] With Riccati ODE standard form \[ y' = f_0(x)+ f_1(x)y+f_2(x)y^{2} \] Shows that \(f_0(x)=-\frac {b \,x^{\alpha }+c}{x^{2}}\), \(f_1(x)=0\) 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 &=0\\ f_2^2 f_0 &=-\frac {a^{2} \left (b \,x^{\alpha }+c \right )}{x^{2}} \end {align*}
Substituting the above terms back in equation (2) gives \begin {align*} -a u^{\prime \prime }\left (x \right )-\frac {a^{2} \left (b \,x^{\alpha }+c \right ) u \left (x \right )}{x^{2}} &=0 \end {align*}
Solving the above ODE (this ode solved using Maple, not this program), gives
\[ u \left (x \right ) = \left (\operatorname {BesselJ}\left (\frac {\sqrt {-4 a c +1}}{\alpha }, \frac {2 \sqrt {b a}\, x^{\frac {\alpha }{2}}}{\alpha }\right ) c_{1} +\operatorname {BesselY}\left (\frac {\sqrt {-4 a c +1}}{\alpha }, \frac {2 \sqrt {b a}\, x^{\frac {\alpha }{2}}}{\alpha }\right ) c_{2} \right ) \sqrt {x} \] The above shows that \[ u^{\prime }\left (x \right ) = \frac {-2 \sqrt {b a}\, \left (\operatorname {BesselJ}\left (\frac {\sqrt {-4 a c +1}+\alpha }{\alpha }, \frac {2 \sqrt {b a}\, x^{\frac {\alpha }{2}}}{\alpha }\right ) c_{1} +\operatorname {BesselY}\left (\frac {\sqrt {-4 a c +1}+\alpha }{\alpha }, \frac {2 \sqrt {b a}\, x^{\frac {\alpha }{2}}}{\alpha }\right ) c_{2} \right ) x^{\frac {\alpha }{2}}+\left (\operatorname {BesselJ}\left (\frac {\sqrt {-4 a c +1}}{\alpha }, \frac {2 \sqrt {b a}\, x^{\frac {\alpha }{2}}}{\alpha }\right ) c_{1} +\operatorname {BesselY}\left (\frac {\sqrt {-4 a c +1}}{\alpha }, \frac {2 \sqrt {b a}\, x^{\frac {\alpha }{2}}}{\alpha }\right ) c_{2} \right ) \left (\sqrt {-4 a c +1}+1\right )}{2 \sqrt {x}} \] Using the above in (1) gives the solution \[ y = \frac {-2 \sqrt {b a}\, \left (\operatorname {BesselJ}\left (\frac {\sqrt {-4 a c +1}+\alpha }{\alpha }, \frac {2 \sqrt {b a}\, x^{\frac {\alpha }{2}}}{\alpha }\right ) c_{1} +\operatorname {BesselY}\left (\frac {\sqrt {-4 a c +1}+\alpha }{\alpha }, \frac {2 \sqrt {b a}\, x^{\frac {\alpha }{2}}}{\alpha }\right ) c_{2} \right ) x^{\frac {\alpha }{2}}+\left (\operatorname {BesselJ}\left (\frac {\sqrt {-4 a c +1}}{\alpha }, \frac {2 \sqrt {b a}\, x^{\frac {\alpha }{2}}}{\alpha }\right ) c_{1} +\operatorname {BesselY}\left (\frac {\sqrt {-4 a c +1}}{\alpha }, \frac {2 \sqrt {b a}\, x^{\frac {\alpha }{2}}}{\alpha }\right ) c_{2} \right ) \left (\sqrt {-4 a c +1}+1\right )}{2 x a \left (\operatorname {BesselJ}\left (\frac {\sqrt {-4 a c +1}}{\alpha }, \frac {2 \sqrt {b a}\, x^{\frac {\alpha }{2}}}{\alpha }\right ) c_{1} +\operatorname {BesselY}\left (\frac {\sqrt {-4 a c +1}}{\alpha }, \frac {2 \sqrt {b a}\, x^{\frac {\alpha }{2}}}{\alpha }\right ) c_{2} \right )} \] Dividing both numerator and denominator by \(c_{1}\) gives, after renaming the constant \(\frac {c_{2}}{c_{1}}=c_{3}\) the following solution
\[ y = \frac {-2 \sqrt {b a}\, \left (\operatorname {BesselJ}\left (\frac {\sqrt {-4 a c +1}}{\alpha }+1, \frac {2 \sqrt {b a}\, x^{\frac {\alpha }{2}}}{\alpha }\right ) c_{3} +\operatorname {BesselY}\left (\frac {\sqrt {-4 a c +1}}{\alpha }+1, \frac {2 \sqrt {b a}\, x^{\frac {\alpha }{2}}}{\alpha }\right )\right ) x^{\frac {\alpha }{2}}+\left (\operatorname {BesselJ}\left (\frac {\sqrt {-4 a c +1}}{\alpha }, \frac {2 \sqrt {b a}\, x^{\frac {\alpha }{2}}}{\alpha }\right ) c_{3} +\operatorname {BesselY}\left (\frac {\sqrt {-4 a c +1}}{\alpha }, \frac {2 \sqrt {b a}\, x^{\frac {\alpha }{2}}}{\alpha }\right )\right ) \left (\sqrt {-4 a c +1}+1\right )}{2 x a \left (\operatorname {BesselJ}\left (\frac {\sqrt {-4 a c +1}}{\alpha }, \frac {2 \sqrt {b a}\, x^{\frac {\alpha }{2}}}{\alpha }\right ) c_{3} +\operatorname {BesselY}\left (\frac {\sqrt {-4 a c +1}}{\alpha }, \frac {2 \sqrt {b a}\, x^{\frac {\alpha }{2}}}{\alpha }\right )\right )} \]
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {-2 \sqrt {b a}\, \left (\operatorname {BesselJ}\left (\frac {\sqrt {-4 a c +1}}{\alpha }+1, \frac {2 \sqrt {b a}\, x^{\frac {\alpha }{2}}}{\alpha }\right ) c_{3} +\operatorname {BesselY}\left (\frac {\sqrt {-4 a c +1}}{\alpha }+1, \frac {2 \sqrt {b a}\, x^{\frac {\alpha }{2}}}{\alpha }\right )\right ) x^{\frac {\alpha }{2}}+\left (\operatorname {BesselJ}\left (\frac {\sqrt {-4 a c +1}}{\alpha }, \frac {2 \sqrt {b a}\, x^{\frac {\alpha }{2}}}{\alpha }\right ) c_{3} +\operatorname {BesselY}\left (\frac {\sqrt {-4 a c +1}}{\alpha }, \frac {2 \sqrt {b a}\, x^{\frac {\alpha }{2}}}{\alpha }\right )\right ) \left (\sqrt {-4 a c +1}+1\right )}{2 x a \left (\operatorname {BesselJ}\left (\frac {\sqrt {-4 a c +1}}{\alpha }, \frac {2 \sqrt {b a}\, x^{\frac {\alpha }{2}}}{\alpha }\right ) c_{3} +\operatorname {BesselY}\left (\frac {\sqrt {-4 a c +1}}{\alpha }, \frac {2 \sqrt {b a}\, x^{\frac {\alpha }{2}}}{\alpha }\right )\right )} \\ \end{align*}
Verification of solutions
\[ y = \frac {-2 \sqrt {b a}\, \left (\operatorname {BesselJ}\left (\frac {\sqrt {-4 a c +1}}{\alpha }+1, \frac {2 \sqrt {b a}\, x^{\frac {\alpha }{2}}}{\alpha }\right ) c_{3} +\operatorname {BesselY}\left (\frac {\sqrt {-4 a c +1}}{\alpha }+1, \frac {2 \sqrt {b a}\, x^{\frac {\alpha }{2}}}{\alpha }\right )\right ) x^{\frac {\alpha }{2}}+\left (\operatorname {BesselJ}\left (\frac {\sqrt {-4 a c +1}}{\alpha }, \frac {2 \sqrt {b a}\, x^{\frac {\alpha }{2}}}{\alpha }\right ) c_{3} +\operatorname {BesselY}\left (\frac {\sqrt {-4 a c +1}}{\alpha }, \frac {2 \sqrt {b a}\, x^{\frac {\alpha }{2}}}{\alpha }\right )\right ) \left (\sqrt {-4 a c +1}+1\right )}{2 x a \left (\operatorname {BesselJ}\left (\frac {\sqrt {-4 a c +1}}{\alpha }, \frac {2 \sqrt {b a}\, x^{\frac {\alpha }{2}}}{\alpha }\right ) c_{3} +\operatorname {BesselY}\left (\frac {\sqrt {-4 a c +1}}{\alpha }, \frac {2 \sqrt {b a}\, x^{\frac {\alpha }{2}}}{\alpha }\right )\right )} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x^{2} \left (y^{\prime }+a y^{2}\right )=-b \,x^{\alpha }-c \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-\frac {y^{2} a \,x^{2}+b \,x^{\alpha }+c}{x^{2}} \end {array} \]
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 Special trying Riccati sub-methods: trying Riccati_symmetries trying Riccati to 2nd Order -> Calling odsolve with the ODE`, diff(diff(y(x), x), x) = -a*(b*x^(-2+alpha)*x^2+c)*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 -> 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: 0.016 (sec). Leaf size: 219
dsolve(x^2*(diff(y(x),x)+a*y(x)^2) + b*x^alpha + c=0,y(x), singsol=all)
\[ y \left (x \right ) = \frac {-2 \sqrt {a b}\, \left (\operatorname {BesselY}\left (\frac {\sqrt {-4 a c +1}}{\alpha }+1, \frac {2 \sqrt {a b}\, x^{\frac {\alpha }{2}}}{\alpha }\right ) c_{1} +\operatorname {BesselJ}\left (\frac {\sqrt {-4 a c +1}}{\alpha }+1, \frac {2 \sqrt {a b}\, x^{\frac {\alpha }{2}}}{\alpha }\right )\right ) x^{\frac {\alpha }{2}}+\left (\sqrt {-4 a c +1}+1\right ) \left (\operatorname {BesselY}\left (\frac {\sqrt {-4 a c +1}}{\alpha }, \frac {2 \sqrt {a b}\, x^{\frac {\alpha }{2}}}{\alpha }\right ) c_{1} +\operatorname {BesselJ}\left (\frac {\sqrt {-4 a c +1}}{\alpha }, \frac {2 \sqrt {a b}\, x^{\frac {\alpha }{2}}}{\alpha }\right )\right )}{2 x a \left (\operatorname {BesselY}\left (\frac {\sqrt {-4 a c +1}}{\alpha }, \frac {2 \sqrt {a b}\, x^{\frac {\alpha }{2}}}{\alpha }\right ) c_{1} +\operatorname {BesselJ}\left (\frac {\sqrt {-4 a c +1}}{\alpha }, \frac {2 \sqrt {a b}\, x^{\frac {\alpha }{2}}}{\alpha }\right )\right )} \]
✓ Solution by Mathematica
Time used: 1.108 (sec). Leaf size: 1777
DSolve[x^2*(y'[x]+a*y[x]^2) + b*x^\[Alpha] + c==0,y[x],x,IncludeSingularSolutions -> True]
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