Internal problem ID [3519]
Internal file name [OUTPUT/3012_Sunday_June_05_2022_08_49_34_AM_9278046/index.tex]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 9
Problem number: 263.
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]
\[ \boxed {x^{2} y^{\prime }-y^{2} x^{2}=a +b \,x^{n}} \]
In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= \frac {a +b \,x^{n}+y^{2} x^{2}}{x^{2}} \end {align*}
This is a Riccati ODE. Comparing the ODE to solve \[ y' = y^{2}+\frac {b \,x^{n}}{x^{2}}+\frac {a}{x^{2}} \] With Riccati ODE standard form \[ y' = f_0(x)+ f_1(x)y+f_2(x)y^{2} \] Shows that \(f_0(x)=\frac {a +b \,x^{n}}{x^{2}}\), \(f_1(x)=0\) and \(f_2(x)=1\). Let \begin {align*} y &= \frac {-u'}{f_2 u} \\ &= \frac {-u'}{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 +b \,x^{n}}{x^{2}} \end {align*}
Substituting the above terms back in equation (2) gives \begin {align*} u^{\prime \prime }\left (x \right )+\frac {\left (a +b \,x^{n}\right ) u \left (x \right )}{x^{2}} &=0 \end {align*}
Solving the above ODE (this ode solved using Maple, not this program), gives
\[ u \left (x \right ) = \left (\operatorname {BesselJ}\left (\frac {\sqrt {1-4 a}}{n}, \frac {2 \sqrt {b}\, x^{\frac {n}{2}}}{n}\right ) c_{1} +\operatorname {BesselY}\left (\frac {\sqrt {1-4 a}}{n}, \frac {2 \sqrt {b}\, x^{\frac {n}{2}}}{n}\right ) c_{2} \right ) \sqrt {x} \] The above shows that \[ u^{\prime }\left (x \right ) = -\frac {\sqrt {b}\, \left (\operatorname {BesselJ}\left (\frac {\sqrt {1-4 a}+n}{n}, \frac {2 \sqrt {b}\, x^{\frac {n}{2}}}{n}\right ) c_{1} +\operatorname {BesselY}\left (\frac {\sqrt {1-4 a}+n}{n}, \frac {2 \sqrt {b}\, x^{\frac {n}{2}}}{n}\right ) c_{2} \right ) x^{\frac {n}{2}}-\frac {\left (\operatorname {BesselJ}\left (\frac {\sqrt {1-4 a}}{n}, \frac {2 \sqrt {b}\, x^{\frac {n}{2}}}{n}\right ) c_{1} +\operatorname {BesselY}\left (\frac {\sqrt {1-4 a}}{n}, \frac {2 \sqrt {b}\, x^{\frac {n}{2}}}{n}\right ) c_{2} \right ) \left (\sqrt {1-4 a}+1\right )}{2}}{\sqrt {x}} \] Using the above in (1) gives the solution \[ y = \frac {\sqrt {b}\, \left (\operatorname {BesselJ}\left (\frac {\sqrt {1-4 a}+n}{n}, \frac {2 \sqrt {b}\, x^{\frac {n}{2}}}{n}\right ) c_{1} +\operatorname {BesselY}\left (\frac {\sqrt {1-4 a}+n}{n}, \frac {2 \sqrt {b}\, x^{\frac {n}{2}}}{n}\right ) c_{2} \right ) x^{\frac {n}{2}}-\frac {\left (\operatorname {BesselJ}\left (\frac {\sqrt {1-4 a}}{n}, \frac {2 \sqrt {b}\, x^{\frac {n}{2}}}{n}\right ) c_{1} +\operatorname {BesselY}\left (\frac {\sqrt {1-4 a}}{n}, \frac {2 \sqrt {b}\, x^{\frac {n}{2}}}{n}\right ) c_{2} \right ) \left (\sqrt {1-4 a}+1\right )}{2}}{x \left (\operatorname {BesselJ}\left (\frac {\sqrt {1-4 a}}{n}, \frac {2 \sqrt {b}\, x^{\frac {n}{2}}}{n}\right ) c_{1} +\operatorname {BesselY}\left (\frac {\sqrt {1-4 a}}{n}, \frac {2 \sqrt {b}\, x^{\frac {n}{2}}}{n}\right ) c_{2} \right )} \] Dividing both numerator and denominator by \(c_{1}\) gives, after renaming the constant \(\frac {c_{2}}{c_{1}}=c_{3}\) the following solution
\[ y = \frac {2 \sqrt {b}\, \left (\operatorname {BesselJ}\left (\frac {\sqrt {1-4 a}}{n}+1, \frac {2 \sqrt {b}\, x^{\frac {n}{2}}}{n}\right ) c_{3} +\operatorname {BesselY}\left (\frac {\sqrt {1-4 a}}{n}+1, \frac {2 \sqrt {b}\, x^{\frac {n}{2}}}{n}\right )\right ) x^{\frac {n}{2}}-\left (\operatorname {BesselJ}\left (\frac {\sqrt {1-4 a}}{n}, \frac {2 \sqrt {b}\, x^{\frac {n}{2}}}{n}\right ) c_{3} +\operatorname {BesselY}\left (\frac {\sqrt {1-4 a}}{n}, \frac {2 \sqrt {b}\, x^{\frac {n}{2}}}{n}\right )\right ) \left (\sqrt {1-4 a}+1\right )}{2 x \left (\operatorname {BesselJ}\left (\frac {\sqrt {1-4 a}}{n}, \frac {2 \sqrt {b}\, x^{\frac {n}{2}}}{n}\right ) c_{3} +\operatorname {BesselY}\left (\frac {\sqrt {1-4 a}}{n}, \frac {2 \sqrt {b}\, x^{\frac {n}{2}}}{n}\right )\right )} \]
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {2 \sqrt {b}\, \left (\operatorname {BesselJ}\left (\frac {\sqrt {1-4 a}}{n}+1, \frac {2 \sqrt {b}\, x^{\frac {n}{2}}}{n}\right ) c_{3} +\operatorname {BesselY}\left (\frac {\sqrt {1-4 a}}{n}+1, \frac {2 \sqrt {b}\, x^{\frac {n}{2}}}{n}\right )\right ) x^{\frac {n}{2}}-\left (\operatorname {BesselJ}\left (\frac {\sqrt {1-4 a}}{n}, \frac {2 \sqrt {b}\, x^{\frac {n}{2}}}{n}\right ) c_{3} +\operatorname {BesselY}\left (\frac {\sqrt {1-4 a}}{n}, \frac {2 \sqrt {b}\, x^{\frac {n}{2}}}{n}\right )\right ) \left (\sqrt {1-4 a}+1\right )}{2 x \left (\operatorname {BesselJ}\left (\frac {\sqrt {1-4 a}}{n}, \frac {2 \sqrt {b}\, x^{\frac {n}{2}}}{n}\right ) c_{3} +\operatorname {BesselY}\left (\frac {\sqrt {1-4 a}}{n}, \frac {2 \sqrt {b}\, x^{\frac {n}{2}}}{n}\right )\right )} \\ \end{align*}
Verification of solutions
\[ y = \frac {2 \sqrt {b}\, \left (\operatorname {BesselJ}\left (\frac {\sqrt {1-4 a}}{n}+1, \frac {2 \sqrt {b}\, x^{\frac {n}{2}}}{n}\right ) c_{3} +\operatorname {BesselY}\left (\frac {\sqrt {1-4 a}}{n}+1, \frac {2 \sqrt {b}\, x^{\frac {n}{2}}}{n}\right )\right ) x^{\frac {n}{2}}-\left (\operatorname {BesselJ}\left (\frac {\sqrt {1-4 a}}{n}, \frac {2 \sqrt {b}\, x^{\frac {n}{2}}}{n}\right ) c_{3} +\operatorname {BesselY}\left (\frac {\sqrt {1-4 a}}{n}, \frac {2 \sqrt {b}\, x^{\frac {n}{2}}}{n}\right )\right ) \left (\sqrt {1-4 a}+1\right )}{2 x \left (\operatorname {BesselJ}\left (\frac {\sqrt {1-4 a}}{n}, \frac {2 \sqrt {b}\, x^{\frac {n}{2}}}{n}\right ) c_{3} +\operatorname {BesselY}\left (\frac {\sqrt {1-4 a}}{n}, \frac {2 \sqrt {b}\, x^{\frac {n}{2}}}{n}\right )\right )} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x^{2} y^{\prime }-y^{2} x^{2}=a +b \,x^{n} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {a +b \,x^{n}+y^{2} x^{2}}{x^{2}} \end {array} \]
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 Special trying Riccati sub-methods: trying Riccati_symmetries trying Riccati to 2nd Order -> Calling odsolve with the ODE`, diff(diff(y(x), x), x) = -(x^(n-2)*b*x^2+a)*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: 0.0 (sec). Leaf size: 196
dsolve(x^2*diff(y(x),x) = a+b*x^n+x^2*y(x)^2,y(x), singsol=all)
\[ y \left (x \right ) = \frac {2 \sqrt {b}\, \left (\operatorname {BesselY}\left (\frac {\sqrt {1-4 a}}{n}+1, \frac {2 \sqrt {b}\, x^{\frac {n}{2}}}{n}\right ) c_{1} +\operatorname {BesselJ}\left (\frac {\sqrt {1-4 a}}{n}+1, \frac {2 \sqrt {b}\, x^{\frac {n}{2}}}{n}\right )\right ) x^{\frac {n}{2}}-\left (\sqrt {1-4 a}+1\right ) \left (\operatorname {BesselY}\left (\frac {\sqrt {1-4 a}}{n}, \frac {2 \sqrt {b}\, x^{\frac {n}{2}}}{n}\right ) c_{1} +\operatorname {BesselJ}\left (\frac {\sqrt {1-4 a}}{n}, \frac {2 \sqrt {b}\, x^{\frac {n}{2}}}{n}\right )\right )}{2 x \left (\operatorname {BesselY}\left (\frac {\sqrt {1-4 a}}{n}, \frac {2 \sqrt {b}\, x^{\frac {n}{2}}}{n}\right ) c_{1} +\operatorname {BesselJ}\left (\frac {\sqrt {1-4 a}}{n}, \frac {2 \sqrt {b}\, x^{\frac {n}{2}}}{n}\right )\right )} \]
✓ Solution by Mathematica
Time used: 0.976 (sec). Leaf size: 1434
DSolve[x^2 y'[x]==a+b x^n + x^2 y[x]^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {-n^{\frac {2 \sqrt {(1-4 a) n^2}}{n^2}+1} \left (x^n\right )^{\frac {i \sqrt {4 a-1}}{n}+1} \operatorname {BesselJ}\left (\frac {\sqrt {(1-4 a) n^2}}{n^2}-1,\frac {2 \sqrt {b} \sqrt {x^n}}{n}\right ) \operatorname {Gamma}\left (\frac {n+\sqrt {1-4 a}}{n}\right ) b^{\frac {i \sqrt {4 a-1}}{n}+\frac {1}{2}}+n^{\frac {2 \sqrt {(1-4 a) n^2}}{n^2}+1} \left (x^n\right )^{\frac {i \sqrt {4 a-1}}{n}+1} \operatorname {BesselJ}\left (\frac {\sqrt {(1-4 a) n^2}}{n^2}+1,\frac {2 \sqrt {b} \sqrt {x^n}}{n}\right ) \operatorname {Gamma}\left (\frac {n+\sqrt {1-4 a}}{n}\right ) b^{\frac {i \sqrt {4 a-1}}{n}+\frac {1}{2}}-i \sqrt {4 a-1} n^{\frac {2 \sqrt {(1-4 a) n^2}}{n^2}+1} \left (x^n\right )^{\frac {i \sqrt {4 a-1}}{n}+\frac {1}{2}} \operatorname {BesselJ}\left (\frac {\sqrt {(1-4 a) n^2}}{n^2},\frac {2 \sqrt {b} \sqrt {x^n}}{n}\right ) \operatorname {Gamma}\left (\frac {n+\sqrt {1-4 a}}{n}\right ) b^{\frac {i \sqrt {4 a-1}}{n}}-n^{\frac {2 \sqrt {(1-4 a) n^2}}{n^2}+1} \left (x^n\right )^{\frac {i \sqrt {4 a-1}}{n}+\frac {1}{2}} \operatorname {BesselJ}\left (\frac {\sqrt {(1-4 a) n^2}}{n^2},\frac {2 \sqrt {b} \sqrt {x^n}}{n}\right ) \operatorname {Gamma}\left (\frac {n+\sqrt {1-4 a}}{n}\right ) b^{\frac {i \sqrt {4 a-1}}{n}}+n^{\frac {2 \sqrt {(1-4 a) n^2}}{n^2}} \sqrt {(1-4 a) n^2} \left (x^n\right )^{\frac {i \sqrt {4 a-1}}{n}+\frac {1}{2}} \operatorname {BesselJ}\left (\frac {\sqrt {(1-4 a) n^2}}{n^2},\frac {2 \sqrt {b} \sqrt {x^n}}{n}\right ) \operatorname {Gamma}\left (\frac {n+\sqrt {1-4 a}}{n}\right ) b^{\frac {i \sqrt {4 a-1}}{n}}-n^{\frac {2 i \sqrt {4 a-1}}{n}} \left (-i \sqrt {4 a-1} n+n+\sqrt {(1-4 a) n^2}\right ) \left (x^n\right )^{\frac {\sqrt {(1-4 a) n^2}}{n^2}+\frac {1}{2}} \operatorname {BesselJ}\left (-\frac {\sqrt {(1-4 a) n^2}}{n^2},\frac {2 \sqrt {b} \sqrt {x^n}}{n}\right ) c_1 \operatorname {Gamma}\left (1-\frac {\sqrt {1-4 a}}{n}\right ) b^{\frac {\sqrt {(1-4 a) n^2}}{n^2}}-n^{\frac {2 i \sqrt {4 a-1}}{n}+1} \left (x^n\right )^{\frac {\sqrt {(1-4 a) n^2}}{n^2}+1} \operatorname {BesselJ}\left (-\frac {\sqrt {(1-4 a) n^2}}{n^2}-1,\frac {2 \sqrt {b} \sqrt {x^n}}{n}\right ) c_1 \operatorname {Gamma}\left (1-\frac {\sqrt {1-4 a}}{n}\right ) b^{\frac {\sqrt {(1-4 a) n^2}}{n^2}+\frac {1}{2}}+n^{\frac {2 i \sqrt {4 a-1}}{n}+1} \left (x^n\right )^{\frac {\sqrt {(1-4 a) n^2}}{n^2}+1} \operatorname {BesselJ}\left (1-\frac {\sqrt {(1-4 a) n^2}}{n^2},\frac {2 \sqrt {b} \sqrt {x^n}}{n}\right ) c_1 \operatorname {Gamma}\left (1-\frac {\sqrt {1-4 a}}{n}\right ) b^{\frac {\sqrt {(1-4 a) n^2}}{n^2}+\frac {1}{2}}}{2 n x \sqrt {x^n} \left (b^{\frac {i \sqrt {4 a-1}}{n}} n^{\frac {2 \sqrt {(1-4 a) n^2}}{n^2}} \operatorname {BesselJ}\left (\frac {\sqrt {(1-4 a) n^2}}{n^2},\frac {2 \sqrt {b} \sqrt {x^n}}{n}\right ) \operatorname {Gamma}\left (\frac {n+\sqrt {1-4 a}}{n}\right ) \left (x^n\right )^{\frac {i \sqrt {4 a-1}}{n}}+b^{\frac {\sqrt {(1-4 a) n^2}}{n^2}} n^{\frac {2 i \sqrt {4 a-1}}{n}} \operatorname {BesselJ}\left (-\frac {\sqrt {(1-4 a) n^2}}{n^2},\frac {2 \sqrt {b} \sqrt {x^n}}{n}\right ) c_1 \operatorname {Gamma}\left (1-\frac {\sqrt {1-4 a}}{n}\right ) \left (x^n\right )^{\frac {\sqrt {(1-4 a) n^2}}{n^2}}\right )} \\ y(x)\to \frac {\frac {\sqrt {b} \sqrt {x^n} \left (\operatorname {BesselJ}\left (1-\frac {\sqrt {(1-4 a) n^2}}{n^2},\frac {2 \sqrt {b} \sqrt {x^n}}{n}\right )-\operatorname {BesselJ}\left (-\frac {\sqrt {(1-4 a) n^2}}{n^2}-1,\frac {2 \sqrt {b} \sqrt {x^n}}{n}\right )\right )}{\operatorname {BesselJ}\left (-\frac {\sqrt {(1-4 a) n^2}}{n^2},\frac {2 \sqrt {b} \sqrt {x^n}}{n}\right )}-\frac {\sqrt {(1-4 a) n^2}}{n}+i \sqrt {4 a-1}-1}{2 x} \\ \end{align*}