Internal problem ID [3424]
Internal file name [OUTPUT/2917_Sunday_June_05_2022_08_47_02_AM_35642763/index.tex]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 6
Problem number: 168.
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]
\[ \boxed {x y^{\prime }-\left (n +y b \right ) y=a \,x^{2 n}} \]
In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= \frac {b \,y^{2}+a \,x^{2 n}+n y}{x} \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 \(f_0(x)=\frac {a \,x^{2 n}}{x}\), \(f_1(x)=\frac {n}{x}\) and \(f_2(x)=\frac {b}{x}\). Let \begin {align*} y &= \frac {-u'}{f_2 u} \\ &= \frac {-u'}{\frac {b 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 {b}{x^{2}}\\ f_1 f_2 &=\frac {n b}{x^{2}}\\ f_2^2 f_0 &=\frac {b^{2} a \,x^{2 n}}{x^{3}} \end {align*}
Substituting the above terms back in equation (2) gives \begin {align*} \frac {b u^{\prime \prime }\left (x \right )}{x}-\left (-\frac {b}{x^{2}}+\frac {n b}{x^{2}}\right ) u^{\prime }\left (x \right )+\frac {b^{2} a \,x^{2 n} u \left (x \right )}{x^{3}} &=0 \end {align*}
Solving the above ODE (this ode solved using Maple, not this program), gives
\[ u \left (x \right ) = c_{1} \sin \left (\frac {x^{n} \sqrt {a b}}{n}\right )+c_{2} \cos \left (\frac {x^{n} \sqrt {a b}}{n}\right ) \] 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 \[ y = -\frac {x^{n} \sqrt {a b}\, \left (c_{1} \cos \left (\frac {x^{n} \sqrt {a b}}{n}\right )-c_{2} \sin \left (\frac {x^{n} \sqrt {a b}}{n}\right )\right )}{b \left (c_{1} \sin \left (\frac {x^{n} \sqrt {a b}}{n}\right )+c_{2} \cos \left (\frac {x^{n} \sqrt {a b}}{n}\right )\right )} \] Dividing both numerator and denominator by \(c_{1}\) gives, after renaming the constant \(\frac {c_{2}}{c_{1}}=c_{3}\) the following solution
\[ y = \frac {\left (-c_{3} \cos \left (\frac {x^{n} \sqrt {a b}}{n}\right )+\sin \left (\frac {x^{n} \sqrt {a b}}{n}\right )\right ) x^{n} \sqrt {a b}}{\left (c_{3} \sin \left (\frac {x^{n} \sqrt {a b}}{n}\right )+\cos \left (\frac {x^{n} \sqrt {a b}}{n}\right )\right ) b} \]
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {\left (-c_{3} \cos \left (\frac {x^{n} \sqrt {a b}}{n}\right )+\sin \left (\frac {x^{n} \sqrt {a b}}{n}\right )\right ) x^{n} \sqrt {a 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 ) b} \\ \end{align*}
Verification of solutions
\[ y = \frac {\left (-c_{3} \cos \left (\frac {x^{n} \sqrt {a b}}{n}\right )+\sin \left (\frac {x^{n} \sqrt {a b}}{n}\right )\right ) x^{n} \sqrt {a b}}{\left (c_{3} \sin \left (\frac {x^{n} \sqrt {a b}}{n}\right )+\cos \left (\frac {x^{n} \sqrt {a b}}{n}\right )\right ) b} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x y^{\prime }-\left (n +y b \right ) y=a \,x^{2 n} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {a \,x^{2 n}+\left (n +y b \right ) y}{x} \end {array} \]
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 <- Chini successful`
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 34
dsolve(x*diff(y(x),x) = a*x^(2*n)+(n+b*y(x))*y(x),y(x), singsol=all)
\[ y \left (x \right ) = \frac {\tan \left (\frac {x^{n} \sqrt {a}\, \sqrt {b}-c_{1} n}{n}\right ) \sqrt {a}\, x^{n}}{\sqrt {b}} \]
✓ Solution by Mathematica
Time used: 0.336 (sec). Leaf size: 139
DSolve[x y'[x]==a x^(2 n)+(n+b y[x])y[x],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*}