Internal problem ID [10369]
Internal file name [OUTPUT/9317_Monday_June_06_2022_01_51_20_PM_83489069/index.tex]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev.
Second edition
Section: Chapter 1, section 1.2. Riccati Equation. 1.2.2. Equations Containing Power
Functions
Problem number: 40.
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]
\[ \boxed {y^{\prime } x -a \,x^{n} y^{2}-m y=-a \,b^{2} x^{n +2 m}} \]
In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= \frac {a \,x^{n} y^{2}+m y -a \,b^{2} x^{n +2 m}}{x} \end {align*}
This is a Riccati ODE. Comparing the ODE to solve \[ y' = \frac {a \,x^{n} y^{2}}{x}+\frac {m y}{x}-\frac {a \,b^{2} x^{n} x^{2 m}}{x} \] With Riccati ODE standard form \[ y' = f_0(x)+ f_1(x)y+f_2(x)y^{2} \] Shows that \(f_0(x)=-\frac {a \,b^{2} x^{n +2 m}}{x}\), \(f_1(x)=\frac {m}{x}\) and \(f_2(x)=\frac {a \,x^{n}}{x}\). Let \begin {align*} y &= \frac {-u'}{f_2 u} \\ &= \frac {-u'}{\frac {a \,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 {a \,x^{n}}{x^{2}}+\frac {a \,x^{n} n}{x^{2}}\\ f_1 f_2 &=\frac {m a \,x^{n}}{x^{2}}\\ f_2^2 f_0 &=-\frac {a^{3} x^{2 n} b^{2} x^{n +2 m}}{x^{3}} \end {align*}
Substituting the above terms back in equation (2) gives \begin {align*} \frac {a \,x^{n} u^{\prime \prime }\left (x \right )}{x}-\left (-\frac {a \,x^{n}}{x^{2}}+\frac {a \,x^{n} n}{x^{2}}+\frac {m a \,x^{n}}{x^{2}}\right ) u^{\prime }\left (x \right )-\frac {a^{3} x^{2 n} b^{2} x^{n +2 m} u \left (x \right )}{x^{3}} &=0 \end {align*}
Solving the above ODE (this ode solved using Maple, not this program), gives
\[ u \left (x \right ) = c_{1} \sinh \left (\frac {a b \,x^{m +n}}{m +n}\right )+c_{2} \cosh \left (\frac {a b \,x^{m +n}}{m +n}\right ) \] The above shows that \[ u^{\prime }\left (x \right ) = a b \,x^{m +n -1} \left (c_{1} \cosh \left (\frac {a b \,x^{m +n}}{m +n}\right )+c_{2} \sinh \left (\frac {a b \,x^{m +n}}{m +n}\right )\right ) \] Using the above in (1) gives the solution \[ y = -\frac {b \,x^{m +n -1} \left (c_{1} \cosh \left (\frac {a b \,x^{m +n}}{m +n}\right )+c_{2} \sinh \left (\frac {a b \,x^{m +n}}{m +n}\right )\right ) x \,x^{-n}}{c_{1} \sinh \left (\frac {a b \,x^{m +n}}{m +n}\right )+c_{2} \cosh \left (\frac {a b \,x^{m +n}}{m +n}\right )} \] Dividing both numerator and denominator by \(c_{1}\) gives, after renaming the constant \(\frac {c_{2}}{c_{1}}=c_{3}\) the following solution
\[ y = -\frac {b \,x^{m} \left (c_{3} \cosh \left (\frac {a b \,x^{m +n}}{m +n}\right )+\sinh \left (\frac {a b \,x^{m +n}}{m +n}\right )\right )}{c_{3} \sinh \left (\frac {a b \,x^{m +n}}{m +n}\right )+\cosh \left (\frac {a b \,x^{m +n}}{m +n}\right )} \]
The solution(s) found are the following \begin{align*} \tag{1} y &= -\frac {b \,x^{m} \left (c_{3} \cosh \left (\frac {a b \,x^{m +n}}{m +n}\right )+\sinh \left (\frac {a b \,x^{m +n}}{m +n}\right )\right )}{c_{3} \sinh \left (\frac {a b \,x^{m +n}}{m +n}\right )+\cosh \left (\frac {a b \,x^{m +n}}{m +n}\right )} \\ \end{align*}
Verification of solutions
\[ y = -\frac {b \,x^{m} \left (c_{3} \cosh \left (\frac {a b \,x^{m +n}}{m +n}\right )+\sinh \left (\frac {a b \,x^{m +n}}{m +n}\right )\right )}{c_{3} \sinh \left (\frac {a b \,x^{m +n}}{m +n}\right )+\cosh \left (\frac {a b \,x^{m +n}}{m +n}\right )} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime } x -a \,x^{n} y^{2}-m y=-a \,b^{2} x^{n +2 m} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {a \,x^{n} y^{2}+m y-a \,b^{2} x^{n +2 m}}{x} \end {array} \]
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: 0.016 (sec). Leaf size: 34
dsolve(x*diff(y(x),x)=a*x^n*y(x)^2+m*y(x)-a*b^2*x^(n+2*m),y(x), singsol=all)
\[ y \left (x \right ) = i \tan \left (\frac {c_{1} \left (n +m \right )+i a b \,x^{n +m}}{n +m}\right ) b \,x^{m} \]
✓ Solution by Mathematica
Time used: 1.736 (sec). Leaf size: 43
DSolve[x*y'[x]==a*x^n*y[x]^2+m*y[x]-a*b^2*x^(n+2*m),y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \sqrt {-b^2} x^m \tan \left (\frac {a \sqrt {-b^2} x^{m+n}}{m+n}+c_1\right ) \]