Internal problem ID [10532]
Internal file name [OUTPUT/9480_Monday_June_06_2022_02_48_23_PM_47763726/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. subsection 1.2.6-3. Equations with
tangent.
Problem number: 35.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "riccati"
Maple gives the following as the ode type
[[_1st_order, `_with_symmetry_[F(x),G(x)]`], _Riccati]
Unable to solve or complete the solution.
\[ \boxed {y^{\prime }-a \tan \left (\lambda x +\mu \right )^{k} \left (y-b \,x^{n}-c \right )^{2}=b n \,x^{n -1}} \]
In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= x^{2 n} \tan \left (\lambda x +\mu \right )^{k} a \,b^{2}+2 x^{n} \tan \left (\lambda x +\mu \right )^{k} a b c -2 x^{n} \tan \left (\lambda x +\mu \right )^{k} a b y +\tan \left (\lambda x +\mu \right )^{k} a \,c^{2}-2 \tan \left (\lambda x +\mu \right )^{k} a c y +\tan \left (\lambda x +\mu \right )^{k} a \,y^{2}+b n \,x^{n -1} \end {align*}
This is a Riccati ODE. Comparing the ODE to solve \[ y' = x^{2 n} \left (\frac {\tan \left (\mu \right )}{1-\tan \left (\mu \right ) \tan \left (\lambda x \right )}+\frac {\tan \left (\lambda x \right )}{1-\tan \left (\mu \right ) \tan \left (\lambda x \right )}\right )^{k} a \,b^{2}+2 x^{n} \left (\frac {\tan \left (\mu \right )}{1-\tan \left (\mu \right ) \tan \left (\lambda x \right )}+\frac {\tan \left (\lambda x \right )}{1-\tan \left (\mu \right ) \tan \left (\lambda x \right )}\right )^{k} a b c -2 x^{n} \left (\frac {\tan \left (\mu \right )}{1-\tan \left (\mu \right ) \tan \left (\lambda x \right )}+\frac {\tan \left (\lambda x \right )}{1-\tan \left (\mu \right ) \tan \left (\lambda x \right )}\right )^{k} a b y +\left (\frac {\tan \left (\mu \right )}{1-\tan \left (\mu \right ) \tan \left (\lambda x \right )}+\frac {\tan \left (\lambda x \right )}{1-\tan \left (\mu \right ) \tan \left (\lambda x \right )}\right )^{k} a \,c^{2}-2 \left (\frac {\tan \left (\mu \right )}{1-\tan \left (\mu \right ) \tan \left (\lambda x \right )}+\frac {\tan \left (\lambda x \right )}{1-\tan \left (\mu \right ) \tan \left (\lambda x \right )}\right )^{k} a c y +\left (\frac {\tan \left (\mu \right )}{1-\tan \left (\mu \right ) \tan \left (\lambda x \right )}+\frac {\tan \left (\lambda x \right )}{1-\tan \left (\mu \right ) \tan \left (\lambda x \right )}\right )^{k} a \,y^{2}+\frac {b n \,x^{n}}{x} \] With Riccati ODE standard form \[ y' = f_0(x)+ f_1(x)y+f_2(x)y^{2} \] Shows that \(f_0(x)=x^{2 n} \tan \left (\lambda x +\mu \right )^{k} a \,b^{2}+2 x^{n} \tan \left (\lambda x +\mu \right )^{k} a b c +\tan \left (\lambda x +\mu \right )^{k} a \,c^{2}+b n \,x^{n -1}\), \(f_1(x)=-2 x^{n} \tan \left (\lambda x +\mu \right )^{k} a b -2 a c \tan \left (\lambda x +\mu \right )^{k}\) and \(f_2(x)=a \tan \left (\lambda x +\mu \right )^{k}\). Let \begin {align*} y &= \frac {-u'}{f_2 u} \\ &= \frac {-u'}{a \tan \left (\lambda x +\mu \right )^{k} 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 {a \tan \left (\lambda x +\mu \right )^{k} k \lambda \left (1+\tan \left (\lambda x +\mu \right )^{2}\right )}{\tan \left (\lambda x +\mu \right )}\\ f_1 f_2 &=\left (-2 x^{n} \tan \left (\lambda x +\mu \right )^{k} a b -2 a c \tan \left (\lambda x +\mu \right )^{k}\right ) a \tan \left (\lambda x +\mu \right )^{k}\\ f_2^2 f_0 &=a^{2} \tan \left (\lambda x +\mu \right )^{2 k} \left (x^{2 n} \tan \left (\lambda x +\mu \right )^{k} a \,b^{2}+2 x^{n} \tan \left (\lambda x +\mu \right )^{k} a b c +\tan \left (\lambda x +\mu \right )^{k} a \,c^{2}+b n \,x^{n -1}\right ) \end {align*}
Substituting the above terms back in equation (2) gives \begin {align*} a \tan \left (\lambda x +\mu \right )^{k} u^{\prime \prime }\left (x \right )-\left (\frac {a \tan \left (\lambda x +\mu \right )^{k} k \lambda \left (1+\tan \left (\lambda x +\mu \right )^{2}\right )}{\tan \left (\lambda x +\mu \right )}+\left (-2 x^{n} \tan \left (\lambda x +\mu \right )^{k} a b -2 a c \tan \left (\lambda x +\mu \right )^{k}\right ) a \tan \left (\lambda x +\mu \right )^{k}\right ) u^{\prime }\left (x \right )+a^{2} \tan \left (\lambda x +\mu \right )^{2 k} \left (x^{2 n} \tan \left (\lambda x +\mu \right )^{k} a \,b^{2}+2 x^{n} \tan \left (\lambda x +\mu \right )^{k} a b c +\tan \left (\lambda x +\mu \right )^{k} a \,c^{2}+b n \,x^{n -1}\right ) u \left (x \right ) &=0 \end {align*}
Solving the above ODE (this ode solved using Maple, not this program), gives Unable to solve. Terminating.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-a \tan \left (\lambda x +\mu \right )^{k} \left (y-b \,x^{n}-c \right )^{2}=b n \,x^{n -1} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=a \tan \left (\lambda x +\mu \right )^{k} \left (y-b \,x^{n}-c \right )^{2}+b n \,x^{n -1} \end {array} \]
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: <- Riccati particular case Kamke (d) successful`
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 42
dsolve(diff(y(x),x)=a*tan(lambda*x+mu)^k*(y(x)-b*x^n-c)^2+b*n*x^(n-1),y(x), singsol=all)
\[ y \left (x \right ) = b \,x^{n}+c +\frac {1}{c_{1} -a \left (\int \left (-\frac {\tan \left (\mu \right )+\tan \left (x \lambda \right )}{\tan \left (\mu \right ) \tan \left (x \lambda \right )-1}\right )^{k}d x \right )} \]
✓ Solution by Mathematica
Time used: 6.024 (sec). Leaf size: 75
DSolve[y'[x]==a*Tan[\[Lambda]*x+mu]^k*(y[x]-b*x^n-c)^2+b*n*x^(n-1),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{-\frac {a \tan ^{k+1}(\mu +\lambda x) \operatorname {Hypergeometric2F1}\left (1,\frac {k+1}{2},\frac {k+3}{2},-\tan ^2(\mu +x \lambda )\right )}{(k+1) \lambda }+c_1}+b x^n+c \\ y(x)\to b x^n+c \\ \end{align*}