Internal problem ID [3439]
Internal file name [OUTPUT/2932_Sunday_June_05_2022_08_47_23_AM_82395362/index.tex]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 7
Problem number: 183.
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]
\[ \boxed {x y^{\prime }+y b +c \,x^{n} y^{2}=a \,x^{m}} \]
In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= \frac {a \,x^{m}-b y -c \,x^{n} y^{2}}{x} \end {align*}
This is a Riccati ODE. Comparing the ODE to solve \[ y' = -\frac {c \,x^{n} y^{2}}{x}+\frac {a \,x^{m}}{x}-\frac {b y}{x} \] With Riccati ODE standard form \[ y' = f_0(x)+ f_1(x)y+f_2(x)y^{2} \] Shows that \(f_0(x)=\frac {a \,x^{m}}{x}\), \(f_1(x)=-\frac {b}{x}\) and \(f_2(x)=-\frac {c \,x^{n}}{x}\). Let \begin {align*} y &= \frac {-u'}{f_2 u} \\ &= \frac {-u'}{-\frac {c \,x^{n} u}{x}} \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 {c \,x^{n}}{x^{2}}-\frac {c \,x^{n} n}{x^{2}}\\ f_1 f_2 &=\frac {b c \,x^{n}}{x^{2}}\\ f_2^2 f_0 &=\frac {c^{2} x^{2 n} a \,x^{m}}{x^{3}} \end {align*}
Substituting the above terms back in equation (2) gives \begin {align*} -\frac {c \,x^{n} u^{\prime \prime }\left (x \right )}{x}-\left (\frac {c \,x^{n}}{x^{2}}-\frac {c \,x^{n} n}{x^{2}}+\frac {b c \,x^{n}}{x^{2}}\right ) u^{\prime }\left (x \right )+\frac {c^{2} x^{2 n} a \,x^{m} u \left (x \right )}{x^{3}} &=0 \end {align*}
Solving the above ODE (this ode solved using Maple, not this program), gives
\[ u \left (x \right ) = x^{-\frac {b}{2}+\frac {n}{2}} \left (\operatorname {BesselY}\left (\frac {b -n}{m +n}, \frac {2 \sqrt {-a c}\, x^{\frac {m}{2}+\frac {n}{2}}}{m +n}\right ) c_{2} +\operatorname {BesselJ}\left (\frac {b -n}{m +n}, \frac {2 \sqrt {-a c}\, x^{\frac {m}{2}+\frac {n}{2}}}{m +n}\right ) c_{1} \right ) \] The above shows that \[ u^{\prime }\left (x \right ) = x^{-1-\frac {b}{2}+n +\frac {m}{2}} \sqrt {-a c}\, \left (-\operatorname {BesselY}\left (\frac {b +m}{m +n}, \frac {2 \sqrt {-a c}\, x^{\frac {m}{2}+\frac {n}{2}}}{m +n}\right ) c_{2} -\operatorname {BesselJ}\left (\frac {b +m}{m +n}, \frac {2 \sqrt {-a c}\, x^{\frac {m}{2}+\frac {n}{2}}}{m +n}\right ) c_{1} \right ) \] Using the above in (1) gives the solution \[ y = \frac {x^{-1-\frac {b}{2}+n +\frac {m}{2}} \sqrt {-a c}\, \left (-\operatorname {BesselY}\left (\frac {b +m}{m +n}, \frac {2 \sqrt {-a c}\, x^{\frac {m}{2}+\frac {n}{2}}}{m +n}\right ) c_{2} -\operatorname {BesselJ}\left (\frac {b +m}{m +n}, \frac {2 \sqrt {-a c}\, x^{\frac {m}{2}+\frac {n}{2}}}{m +n}\right ) c_{1} \right ) x \,x^{-n} x^{\frac {b}{2}-\frac {n}{2}}}{c \left (\operatorname {BesselY}\left (\frac {b -n}{m +n}, \frac {2 \sqrt {-a c}\, x^{\frac {m}{2}+\frac {n}{2}}}{m +n}\right ) c_{2} +\operatorname {BesselJ}\left (\frac {b -n}{m +n}, \frac {2 \sqrt {-a c}\, x^{\frac {m}{2}+\frac {n}{2}}}{m +n}\right ) c_{1} \right )} \] Dividing both numerator and denominator by \(c_{1}\) gives, after renaming the constant \(\frac {c_{2}}{c_{1}}=c_{3}\) the following solution
\[ y = \frac {x^{\frac {m}{2}-\frac {n}{2}} \sqrt {-a c}\, \left (-\operatorname {BesselY}\left (\frac {b +m}{m +n}, \frac {2 \sqrt {-a c}\, x^{\frac {m}{2}+\frac {n}{2}}}{m +n}\right )-\operatorname {BesselJ}\left (\frac {b +m}{m +n}, \frac {2 \sqrt {-a c}\, x^{\frac {m}{2}+\frac {n}{2}}}{m +n}\right ) c_{3} \right )}{c \left (\operatorname {BesselY}\left (\frac {b -n}{m +n}, \frac {2 \sqrt {-a c}\, x^{\frac {m}{2}+\frac {n}{2}}}{m +n}\right )+\operatorname {BesselJ}\left (\frac {b -n}{m +n}, \frac {2 \sqrt {-a c}\, x^{\frac {m}{2}+\frac {n}{2}}}{m +n}\right ) c_{3} \right )} \]
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {x^{\frac {m}{2}-\frac {n}{2}} \sqrt {-a c}\, \left (-\operatorname {BesselY}\left (\frac {b +m}{m +n}, \frac {2 \sqrt {-a c}\, x^{\frac {m}{2}+\frac {n}{2}}}{m +n}\right )-\operatorname {BesselJ}\left (\frac {b +m}{m +n}, \frac {2 \sqrt {-a c}\, x^{\frac {m}{2}+\frac {n}{2}}}{m +n}\right ) c_{3} \right )}{c \left (\operatorname {BesselY}\left (\frac {b -n}{m +n}, \frac {2 \sqrt {-a c}\, x^{\frac {m}{2}+\frac {n}{2}}}{m +n}\right )+\operatorname {BesselJ}\left (\frac {b -n}{m +n}, \frac {2 \sqrt {-a c}\, x^{\frac {m}{2}+\frac {n}{2}}}{m +n}\right ) c_{3} \right )} \\ \end{align*}
Verification of solutions
\[ y = \frac {x^{\frac {m}{2}-\frac {n}{2}} \sqrt {-a c}\, \left (-\operatorname {BesselY}\left (\frac {b +m}{m +n}, \frac {2 \sqrt {-a c}\, x^{\frac {m}{2}+\frac {n}{2}}}{m +n}\right )-\operatorname {BesselJ}\left (\frac {b +m}{m +n}, \frac {2 \sqrt {-a c}\, x^{\frac {m}{2}+\frac {n}{2}}}{m +n}\right ) c_{3} \right )}{c \left (\operatorname {BesselY}\left (\frac {b -n}{m +n}, \frac {2 \sqrt {-a c}\, x^{\frac {m}{2}+\frac {n}{2}}}{m +n}\right )+\operatorname {BesselJ}\left (\frac {b -n}{m +n}, \frac {2 \sqrt {-a c}\, x^{\frac {m}{2}+\frac {n}{2}}}{m +n}\right ) c_{3} \right )} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x y^{\prime }+y b +c \,x^{n} y^{2}=a \,x^{m} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {a \,x^{m}-y b -c \,x^{n} y^{2}}{x} \end {array} \]
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) = -(b-n+1)*(diff(y(x), x))/x+c*x^(n-1)*x^(m-1)*a*y(x), y(x)` *** Su 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: 166
dsolve(x*diff(y(x),x) = a*x^m-b*y(x)-c*x^n*y(x)^2,y(x), singsol=all)
\[ y \left (x \right ) = \frac {x^{-\frac {n}{2}+\frac {m}{2}} \sqrt {-a c}\, \left (-\operatorname {BesselY}\left (\frac {b +m}{n +m}, \frac {2 \sqrt {-a c}\, x^{\frac {n}{2}+\frac {m}{2}}}{n +m}\right ) c_{1} -\operatorname {BesselJ}\left (\frac {b +m}{n +m}, \frac {2 \sqrt {-a c}\, x^{\frac {n}{2}+\frac {m}{2}}}{n +m}\right )\right )}{c \left (\operatorname {BesselY}\left (\frac {b -n}{n +m}, \frac {2 \sqrt {-a c}\, x^{\frac {n}{2}+\frac {m}{2}}}{n +m}\right ) c_{1} +\operatorname {BesselJ}\left (\frac {b -n}{n +m}, \frac {2 \sqrt {-a c}\, x^{\frac {n}{2}+\frac {m}{2}}}{n +m}\right )\right )} \]
✓ Solution by Mathematica
Time used: 0.973 (sec). Leaf size: 1549
DSolve[x y'[x]==a x^m-b y[x]-c x^n y[x]^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {x^{-n} \left ((-1)^{\frac {n}{m+n}} \sqrt {a} \sqrt {c} m (m+n)^{\frac {2 n}{m+n}} x^{m+n} \operatorname {BesselI}\left (-\frac {b+m}{m+n},\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^{m+n}}}{\sqrt {(m+n)^2}}\right ) \operatorname {Gamma}\left (\frac {-b+m+2 n}{m+n}\right ) \left ((m+n)^2\right )^{\frac {b}{m+n}}+(-1)^{\frac {n}{m+n}} \sqrt {a} \sqrt {c} n (m+n)^{\frac {2 n}{m+n}} x^{m+n} \operatorname {BesselI}\left (-\frac {b+m}{m+n},\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^{m+n}}}{\sqrt {(m+n)^2}}\right ) \operatorname {Gamma}\left (\frac {-b+m+2 n}{m+n}\right ) \left ((m+n)^2\right )^{\frac {b}{m+n}}+(-1)^{\frac {n}{m+n}} \sqrt {a} \sqrt {c} m (m+n)^{\frac {2 n}{m+n}} x^{m+n} \operatorname {BesselI}\left (\frac {n-b}{m+n}+1,\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^{m+n}}}{\sqrt {(m+n)^2}}\right ) \operatorname {Gamma}\left (\frac {-b+m+2 n}{m+n}\right ) \left ((m+n)^2\right )^{\frac {b}{m+n}}+(-1)^{\frac {n}{m+n}} \sqrt {a} \sqrt {c} n (m+n)^{\frac {2 n}{m+n}} x^{m+n} \operatorname {BesselI}\left (\frac {n-b}{m+n}+1,\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^{m+n}}}{\sqrt {(m+n)^2}}\right ) \operatorname {Gamma}\left (\frac {-b+m+2 n}{m+n}\right ) \left ((m+n)^2\right )^{\frac {b}{m+n}}+(-1)^{\frac {b}{m+n}} \sqrt {a} \sqrt {c} (m+n)^{\frac {2 b}{m+n}+1} x^{m+n} \operatorname {BesselI}\left (\frac {b+m}{m+n},\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^{m+n}}}{\sqrt {(m+n)^2}}\right ) c_1 \operatorname {Gamma}\left (\frac {b+m}{m+n}\right ) \left ((m+n)^2\right )^{\frac {n}{m+n}}+(-1)^{\frac {b}{m+n}} \sqrt {a} \sqrt {c} m (m+n)^{\frac {2 b}{m+n}} x^{m+n} \operatorname {BesselI}\left (\frac {b-n}{m+n}-1,\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^{m+n}}}{\sqrt {(m+n)^2}}\right ) c_1 \operatorname {Gamma}\left (\frac {b+m}{m+n}\right ) \left ((m+n)^2\right )^{\frac {n}{m+n}}+(-1)^{\frac {b}{m+n}} \sqrt {a} \sqrt {c} n (m+n)^{\frac {2 b}{m+n}} x^{m+n} \operatorname {BesselI}\left (\frac {b-n}{m+n}-1,\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^{m+n}}}{\sqrt {(m+n)^2}}\right ) c_1 \operatorname {Gamma}\left (\frac {b+m}{m+n}\right ) \left ((m+n)^2\right )^{\frac {n}{m+n}}-(-1)^{\frac {n}{m+n}} b (m+n)^{\frac {2 n}{m+n}} \sqrt {x^{m+n}} \operatorname {BesselI}\left (\frac {n-b}{m+n},\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^{m+n}}}{\sqrt {(m+n)^2}}\right ) \operatorname {Gamma}\left (\frac {-b+m+2 n}{m+n}\right ) \left ((m+n)^2\right )^{\frac {b}{m+n}+\frac {1}{2}}+(-1)^{\frac {n}{m+n}} n (m+n)^{\frac {2 n}{m+n}} \sqrt {x^{m+n}} \operatorname {BesselI}\left (\frac {n-b}{m+n},\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^{m+n}}}{\sqrt {(m+n)^2}}\right ) \operatorname {Gamma}\left (\frac {-b+m+2 n}{m+n}\right ) \left ((m+n)^2\right )^{\frac {b}{m+n}+\frac {1}{2}}+(-1)^{\frac {b}{m+n}} (n-b) (m+n)^{\frac {2 b}{m+n}} \sqrt {x^{m+n}} \operatorname {BesselI}\left (\frac {b-n}{m+n},\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^{m+n}}}{\sqrt {(m+n)^2}}\right ) c_1 \operatorname {Gamma}\left (\frac {b+m}{m+n}\right ) \left ((m+n)^2\right )^{\frac {n}{m+n}+\frac {1}{2}}\right )}{2 c \sqrt {(m+n)^2} \sqrt {x^{m+n}} \left ((-1)^{\frac {n}{m+n}} (m+n)^{\frac {2 n}{m+n}} \operatorname {BesselI}\left (\frac {n-b}{m+n},\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^{m+n}}}{\sqrt {(m+n)^2}}\right ) \operatorname {Gamma}\left (\frac {-b+m+2 n}{m+n}\right ) \left ((m+n)^2\right )^{\frac {b}{m+n}}+(-1)^{\frac {b}{m+n}} (m+n)^{\frac {2 b}{m+n}} \operatorname {BesselI}\left (\frac {b-n}{m+n},\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^{m+n}}}{\sqrt {(m+n)^2}}\right ) c_1 \operatorname {Gamma}\left (\frac {b+m}{m+n}\right ) \left ((m+n)^2\right )^{\frac {n}{m+n}}\right )} \\ y(x)\to \frac {x^{-n} \left (\sqrt {a} \sqrt {c} (m+n) \sqrt {x^{m+n}} \operatorname {BesselI}\left (\frac {b+m}{m+n},\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^{m+n}}}{\sqrt {(m+n)^2}}\right )+(n-b) \sqrt {(m+n)^2} \operatorname {BesselI}\left (\frac {b-n}{m+n},\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^{m+n}}}{\sqrt {(m+n)^2}}\right )+\sqrt {a} \sqrt {c} (m+n) \sqrt {x^{m+n}} \operatorname {BesselI}\left (\frac {b-n}{m+n}-1,\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^{m+n}}}{\sqrt {(m+n)^2}}\right )\right )}{2 c \sqrt {(m+n)^2} \operatorname {BesselI}\left (\frac {b-n}{m+n},\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^{m+n}}}{\sqrt {(m+n)^2}}\right )} \\ \end{align*}