Internal problem ID [3525]
Internal file name [OUTPUT/3018_Sunday_June_05_2022_08_49_45_AM_9252279/index.tex]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 10
Problem number: 269.
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]
\[ \boxed {y^{\prime } x^{2}-b x y-c \,x^{4} y^{2}=a} \]
In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= \frac {c \,x^{4} y^{2}+b x y +a}{x^{2}} \end {align*}
This is a Riccati ODE. Comparing the ODE to solve \[ y' = c \,x^{2} y^{2}+\frac {b y}{x}+\frac {a}{x^{2}} \] With Riccati ODE standard form \[ y' = f_0(x)+ f_1(x)y+f_2(x)y^{2} \] Shows that \(f_0(x)=\frac {a}{x^{2}}\), \(f_1(x)=\frac {b}{x}\) and \(f_2(x)=c \,x^{2}\). Let \begin {align*} y &= \frac {-u'}{f_2 u} \\ &= \frac {-u'}{c \,x^{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' &=2 c x\\ f_1 f_2 &=b c x\\ f_2^2 f_0 &=c^{2} x^{2} a \end {align*}
Substituting the above terms back in equation (2) gives \begin {align*} c \,x^{2} u^{\prime \prime }\left (x \right )-\left (b c x +2 c x \right ) u^{\prime }\left (x \right )+c^{2} x^{2} a u \left (x \right ) &=0 \end {align*}
Solving the above ODE (this ode solved using Maple, not this program), gives
\[ u \left (x \right ) = -\frac {x^{\frac {b}{2}+\frac {1}{2}} \left (x \sqrt {a c}\, \left (\operatorname {BesselJ}\left (\frac {1}{2}-\frac {b}{2}, \sqrt {a c}\, x \right ) c_{1} +\operatorname {BesselY}\left (\frac {1}{2}-\frac {b}{2}, \sqrt {a c}\, x \right ) c_{2} \right )+\left (b +1\right ) \left (\operatorname {BesselJ}\left (-\frac {1}{2}-\frac {b}{2}, \sqrt {a c}\, x \right ) c_{1} +\operatorname {BesselY}\left (-\frac {1}{2}-\frac {b}{2}, \sqrt {a c}\, x \right ) c_{2} \right )\right )}{\sqrt {a c}} \] The above shows that \[ u^{\prime }\left (x \right ) = \frac {a c \,x^{\frac {b}{2}+\frac {3}{2}} \left (-\operatorname {BesselJ}\left (-\frac {1}{2}-\frac {b}{2}, \sqrt {a c}\, x \right ) c_{1} -\operatorname {BesselY}\left (-\frac {1}{2}-\frac {b}{2}, \sqrt {a c}\, x \right ) c_{2} \right )}{\sqrt {a c}} \] Using the above in (1) gives the solution \[ y = \frac {a \,x^{\frac {b}{2}+\frac {3}{2}} \left (-\operatorname {BesselJ}\left (-\frac {1}{2}-\frac {b}{2}, \sqrt {a c}\, x \right ) c_{1} -\operatorname {BesselY}\left (-\frac {1}{2}-\frac {b}{2}, \sqrt {a c}\, x \right ) c_{2} \right ) x^{-\frac {1}{2}-\frac {b}{2}}}{x^{2} \left (x \sqrt {a c}\, \left (\operatorname {BesselJ}\left (\frac {1}{2}-\frac {b}{2}, \sqrt {a c}\, x \right ) c_{1} +\operatorname {BesselY}\left (\frac {1}{2}-\frac {b}{2}, \sqrt {a c}\, x \right ) c_{2} \right )+\left (b +1\right ) \left (\operatorname {BesselJ}\left (-\frac {1}{2}-\frac {b}{2}, \sqrt {a c}\, x \right ) c_{1} +\operatorname {BesselY}\left (-\frac {1}{2}-\frac {b}{2}, \sqrt {a c}\, x \right ) c_{2} \right )\right )} \] Dividing both numerator and denominator by \(c_{1}\) gives, after renaming the constant \(\frac {c_{2}}{c_{1}}=c_{3}\) the following solution
\[ y = -\frac {a \left (\operatorname {BesselJ}\left (-\frac {1}{2}-\frac {b}{2}, \sqrt {a c}\, x \right ) c_{3} +\operatorname {BesselY}\left (-\frac {1}{2}-\frac {b}{2}, \sqrt {a c}\, x \right )\right )}{\left (x \sqrt {a c}\, \left (\operatorname {BesselJ}\left (\frac {1}{2}-\frac {b}{2}, \sqrt {a c}\, x \right ) c_{3} +\operatorname {BesselY}\left (\frac {1}{2}-\frac {b}{2}, \sqrt {a c}\, x \right )\right )+\left (b +1\right ) \left (\operatorname {BesselJ}\left (-\frac {1}{2}-\frac {b}{2}, \sqrt {a c}\, x \right ) c_{3} +\operatorname {BesselY}\left (-\frac {1}{2}-\frac {b}{2}, \sqrt {a c}\, x \right )\right )\right ) x} \]
The solution(s) found are the following \begin{align*} \tag{1} y &= -\frac {a \left (\operatorname {BesselJ}\left (-\frac {1}{2}-\frac {b}{2}, \sqrt {a c}\, x \right ) c_{3} +\operatorname {BesselY}\left (-\frac {1}{2}-\frac {b}{2}, \sqrt {a c}\, x \right )\right )}{\left (x \sqrt {a c}\, \left (\operatorname {BesselJ}\left (\frac {1}{2}-\frac {b}{2}, \sqrt {a c}\, x \right ) c_{3} +\operatorname {BesselY}\left (\frac {1}{2}-\frac {b}{2}, \sqrt {a c}\, x \right )\right )+\left (b +1\right ) \left (\operatorname {BesselJ}\left (-\frac {1}{2}-\frac {b}{2}, \sqrt {a c}\, x \right ) c_{3} +\operatorname {BesselY}\left (-\frac {1}{2}-\frac {b}{2}, \sqrt {a c}\, x \right )\right )\right ) x} \\ \end{align*}
Verification of solutions
\[ y = -\frac {a \left (\operatorname {BesselJ}\left (-\frac {1}{2}-\frac {b}{2}, \sqrt {a c}\, x \right ) c_{3} +\operatorname {BesselY}\left (-\frac {1}{2}-\frac {b}{2}, \sqrt {a c}\, x \right )\right )}{\left (x \sqrt {a c}\, \left (\operatorname {BesselJ}\left (\frac {1}{2}-\frac {b}{2}, \sqrt {a c}\, x \right ) c_{3} +\operatorname {BesselY}\left (\frac {1}{2}-\frac {b}{2}, \sqrt {a c}\, x \right )\right )+\left (b +1\right ) \left (\operatorname {BesselJ}\left (-\frac {1}{2}-\frac {b}{2}, \sqrt {a c}\, x \right ) c_{3} +\operatorname {BesselY}\left (-\frac {1}{2}-\frac {b}{2}, \sqrt {a c}\, x \right )\right )\right ) x} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime } x^{2}-b x y-c \,x^{4} y^{2}=a \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {a +b x y+c \,x^{4} y^{2}}{x^{2}} \end {array} \]
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 to 2nd Order -> Calling odsolve with the ODE`, diff(diff(y(x), x), x) = (2+b)*(diff(y(x), x))/x-y(x)*a*c, 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: 111
dsolve(x^2*diff(y(x),x) = a+b*x*y(x)+c*x^4*y(x)^2,y(x), singsol=all)
\[ y \left (x \right ) = -\frac {a \left (\operatorname {BesselY}\left (-\frac {1}{2}-\frac {b}{2}, x \sqrt {a c}\right ) c_{1} +\operatorname {BesselJ}\left (-\frac {1}{2}-\frac {b}{2}, x \sqrt {a c}\right )\right )}{x \left (x \sqrt {a c}\, \left (c_{1} \operatorname {BesselY}\left (\frac {1}{2}-\frac {b}{2}, x \sqrt {a c}\right )+\operatorname {BesselJ}\left (\frac {1}{2}-\frac {b}{2}, x \sqrt {a c}\right )\right )+\left (b +1\right ) \left (\operatorname {BesselY}\left (-\frac {1}{2}-\frac {b}{2}, x \sqrt {a c}\right ) c_{1} +\operatorname {BesselJ}\left (-\frac {1}{2}-\frac {b}{2}, x \sqrt {a c}\right )\right )\right )} \]
✓ Solution by Mathematica
Time used: 0.387 (sec). Leaf size: 394
DSolve[x^2 y'[x]==a+b x y[x]+c x^4 y[x]^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {\sqrt {a} \sqrt {c} x \operatorname {BesselY}\left (\frac {b+1}{2},\sqrt {a} \sqrt {c} x\right )+(b+3) \operatorname {BesselY}\left (\frac {b+3}{2},\sqrt {a} \sqrt {c} x\right )-\sqrt {a} \sqrt {c} x \operatorname {BesselY}\left (\frac {b+5}{2},\sqrt {a} \sqrt {c} x\right )+\sqrt {a} \sqrt {c} c_1 x \operatorname {BesselJ}\left (\frac {b+1}{2},\sqrt {a} \sqrt {c} x\right )+b c_1 \operatorname {BesselJ}\left (\frac {b+3}{2},\sqrt {a} \sqrt {c} x\right )+3 c_1 \operatorname {BesselJ}\left (\frac {b+3}{2},\sqrt {a} \sqrt {c} x\right )-\sqrt {a} \sqrt {c} c_1 x \operatorname {BesselJ}\left (\frac {b+5}{2},\sqrt {a} \sqrt {c} x\right )}{2 c x^3 \left (\operatorname {BesselY}\left (\frac {b+3}{2},\sqrt {a} \sqrt {c} x\right )+c_1 \operatorname {BesselJ}\left (\frac {b+3}{2},\sqrt {a} \sqrt {c} x\right )\right )} \\ y(x)\to -\frac {\sqrt {a} \sqrt {c} x \operatorname {BesselJ}\left (\frac {b+1}{2},\sqrt {a} \sqrt {c} x\right )+(b+3) \operatorname {BesselJ}\left (\frac {b+3}{2},\sqrt {a} \sqrt {c} x\right )-\sqrt {a} \sqrt {c} x \operatorname {BesselJ}\left (\frac {b+5}{2},\sqrt {a} \sqrt {c} x\right )}{2 c x^3 \operatorname {BesselJ}\left (\frac {b+3}{2},\sqrt {a} \sqrt {c} x\right )} \\ \end{align*}