Internal problem ID [3641]
Internal file name [OUTPUT/3134_Sunday_June_05_2022_08_53_17_AM_48636217/index.tex]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 14
Problem number: 386.
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]
\[ \boxed {x^{n} y^{\prime }-x^{n -1} \left (a \,x^{2 n}+y n -b y^{2}\right )=0} \]
In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= x^{n -1} \left (a \,x^{2 n}+n y -b \,y^{2}\right ) x^{-n} \end {align*}
This is a Riccati ODE. Comparing the ODE to solve \[ y' = -\frac {b \,y^{2}}{x}+\frac {a \,x^{2 n}}{x}+\frac {n y}{x} \] With Riccati ODE standard form \[ y' = f_0(x)+ f_1(x)y+f_2(x)y^{2} \] Shows that \(f_0(x)=x^{n -1} x^{-n} a \,x^{2 n}\), \(f_1(x)=n \,x^{n -1} x^{-n}\) and \(f_2(x)=-x^{n -1} x^{-n} b\). Let \begin {align*} y &= \frac {-u'}{f_2 u} \\ &= \frac {-u'}{-x^{n -1} b \,x^{-n} 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^{n -1} \left (n -1\right ) b \,x^{-n}}{x}+\frac {x^{n -1} b \,x^{-n} n}{x}\\ f_1 f_2 &=-n \,x^{-2+2 n} x^{-2 n} b\\ f_2^2 f_0 &=x^{-3+3 n} b^{2} x^{-3 n} a \,x^{2 n} \end {align*}
Substituting the above terms back in equation (2) gives \begin {align*} -x^{n -1} b \,x^{-n} u^{\prime \prime }\left (x \right )-\left (-\frac {x^{n -1} \left (n -1\right ) b \,x^{-n}}{x}+\frac {x^{n -1} b \,x^{-n} n}{x}-n \,x^{-2+2 n} x^{-2 n} b \right ) u^{\prime }\left (x \right )+x^{-3+3 n} b^{2} x^{-3 n} a \,x^{2 n} u \left (x \right ) &=0 \end {align*}
Solving the above ODE (this ode solved using Maple, not this program), gives
\[ u \left (x \right ) = c_{1} \sin \left (\frac {x^{n} \sqrt {-a b}}{n}\right )+c_{2} \cos \left (\frac {x^{n} \sqrt {-a b}}{n}\right ) \] The above shows that \[ u^{\prime }\left (x \right ) = \frac {x^{n} \sqrt {-a b}\, \left (c_{1} \cos \left (\frac {x^{n} \sqrt {-a b}}{n}\right )-c_{2} \sin \left (\frac {x^{n} \sqrt {-a b}}{n}\right )\right )}{x} \] Using the above in (1) gives the solution \[ y = \frac {x^{2 n} \sqrt {-a b}\, \left (c_{1} \cos \left (\frac {x^{n} \sqrt {-a b}}{n}\right )-c_{2} \sin \left (\frac {x^{n} \sqrt {-a b}}{n}\right )\right ) x^{-n +1}}{x b \left (c_{1} \sin \left (\frac {x^{n} \sqrt {-a b}}{n}\right )+c_{2} \cos \left (\frac {x^{n} \sqrt {-a b}}{n}\right )\right )} \] Dividing both numerator and denominator by \(c_{1}\) gives, after renaming the constant \(\frac {c_{2}}{c_{1}}=c_{3}\) the following solution
\[ y = \frac {x^{n} \sqrt {-a b}\, \left (c_{3} \cos \left (\frac {x^{n} \sqrt {-a b}}{n}\right )-\sin \left (\frac {x^{n} \sqrt {-a b}}{n}\right )\right )}{b \left (c_{3} \sin \left (\frac {x^{n} \sqrt {-a b}}{n}\right )+\cos \left (\frac {x^{n} \sqrt {-a b}}{n}\right )\right )} \]
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {x^{n} \sqrt {-a b}\, \left (c_{3} \cos \left (\frac {x^{n} \sqrt {-a b}}{n}\right )-\sin \left (\frac {x^{n} \sqrt {-a b}}{n}\right )\right )}{b \left (c_{3} \sin \left (\frac {x^{n} \sqrt {-a b}}{n}\right )+\cos \left (\frac {x^{n} \sqrt {-a b}}{n}\right )\right )} \\ \end{align*}
Verification of solutions
\[ y = \frac {x^{n} \sqrt {-a b}\, \left (c_{3} \cos \left (\frac {x^{n} \sqrt {-a b}}{n}\right )-\sin \left (\frac {x^{n} \sqrt {-a b}}{n}\right )\right )}{b \left (c_{3} \sin \left (\frac {x^{n} \sqrt {-a b}}{n}\right )+\cos \left (\frac {x^{n} \sqrt {-a b}}{n}\right )\right )} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x^{n} y^{\prime }-x^{n -1} \left (a \,x^{2 n}+y n -b y^{2}\right )=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {x^{n -1} \left (a \,x^{2 n}+y n -b y^{2}\right )}{x^{n}} \end {array} \]
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 <- Chini successful`
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 35
dsolve(x^n*diff(y(x),x) = x^(n-1)*(a*x^(2*n)+n*y(x)-b*y(x)^2),y(x), singsol=all)
\[ y \left (x \right ) = \frac {\tanh \left (\frac {x^{n} \sqrt {a}\, \sqrt {b}+i c_{1} n}{n}\right ) \sqrt {a}\, x^{n}}{\sqrt {b}} \]
✓ Solution by Mathematica
Time used: 0.322 (sec). Leaf size: 153
DSolve[x^n y'[x]==x^(n-1)(a x^(2 n)+n y[x]-b y[x]^2),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\sqrt {a} x^n \left (-\cos \left (\frac {\sqrt {a} \sqrt {-b} x^n}{n}\right )+c_1 \sin \left (\frac {\sqrt {a} \sqrt {-b} x^n}{n}\right )\right )}{\sqrt {-b} \left (\sin \left (\frac {\sqrt {a} \sqrt {-b} x^n}{n}\right )+c_1 \cos \left (\frac {\sqrt {a} \sqrt {-b} x^n}{n}\right )\right )} \\ y(x)\to \frac {\sqrt {a} x^n \tan \left (\frac {\sqrt {a} \sqrt {-b} x^n}{n}\right )}{\sqrt {-b}} \\ \end{align*}