Internal problem ID [10507]
Internal file name [OUTPUT/9455_Monday_June_06_2022_02_37_33_PM_50920785/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 1.2.6-1. Equations with
sine
Problem number: 10.
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]
\[ \boxed {y^{\prime }-a \sin \left (\lambda x +\mu \right )^{k} \left (y-b \,x^{n}-c \right )^{2}=b n \,x^{n -1}} \]
In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= \sin \left (\lambda x +\mu \right )^{k} x^{2 n} a \,b^{2}+2 \sin \left (\lambda x +\mu \right )^{k} x^{n} a b c -2 \sin \left (\lambda x +\mu \right )^{k} x^{n} a b y +\sin \left (\lambda x +\mu \right )^{k} a \,c^{2}-2 \sin \left (\lambda x +\mu \right )^{k} a c y +\sin \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 \[ y' = \left (\sin \left (\lambda x \right ) \cos \left (\mu \right )+\cos \left (\lambda x \right ) \sin \left (\mu \right )\right )^{k} x^{2 n} a \,b^{2}+2 \left (\sin \left (\lambda x \right ) \cos \left (\mu \right )+\cos \left (\lambda x \right ) \sin \left (\mu \right )\right )^{k} x^{n} a b c -2 \left (\sin \left (\lambda x \right ) \cos \left (\mu \right )+\cos \left (\lambda x \right ) \sin \left (\mu \right )\right )^{k} x^{n} a b y +\left (\sin \left (\lambda x \right ) \cos \left (\mu \right )+\cos \left (\lambda x \right ) \sin \left (\mu \right )\right )^{k} a \,c^{2}-2 \left (\sin \left (\lambda x \right ) \cos \left (\mu \right )+\cos \left (\lambda x \right ) \sin \left (\mu \right )\right )^{k} a c y +\left (\sin \left (\lambda x \right ) \cos \left (\mu \right )+\cos \left (\lambda x \right ) \sin \left (\mu \right )\right )^{k} a \,y^{2}+\frac {b n \,x^{n}}{x} \] With Riccati ODE standard form \[ y' = f_0(x)+ f_1(x)y+f_2(x)y^{2} \] Shows that \(f_0(x)=\sin \left (\lambda x +\mu \right )^{k} x^{2 n} a \,b^{2}+2 \sin \left (\lambda x +\mu \right )^{k} x^{n} a b c +\sin \left (\lambda x +\mu \right )^{k} a \,c^{2}+b n \,x^{n -1}\), \(f_1(x)=-2 \sin \left (\lambda x +\mu \right )^{k} x^{n} a b -2 a c \sin \left (\lambda x +\mu \right )^{k}\) and \(f_2(x)=a \sin \left (\lambda x +\mu \right )^{k}\). Let \begin {align*} y &= \frac {-u'}{f_2 u} \\ &= \frac {-u'}{a \sin \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 \sin \left (\lambda x +\mu \right )^{k} k \lambda \cos \left (\lambda x +\mu \right )}{\sin \left (\lambda x +\mu \right )}\\ f_1 f_2 &=\left (-2 \sin \left (\lambda x +\mu \right )^{k} x^{n} a b -2 a c \sin \left (\lambda x +\mu \right )^{k}\right ) a \sin \left (\lambda x +\mu \right )^{k}\\ f_2^2 f_0 &=a^{2} \sin \left (\lambda x +\mu \right )^{2 k} \left (\sin \left (\lambda x +\mu \right )^{k} x^{2 n} a \,b^{2}+2 \sin \left (\lambda x +\mu \right )^{k} x^{n} a b c +\sin \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 \sin \left (\lambda x +\mu \right )^{k} u^{\prime \prime }\left (x \right )-\left (\frac {a \sin \left (\lambda x +\mu \right )^{k} k \lambda \cos \left (\lambda x +\mu \right )}{\sin \left (\lambda x +\mu \right )}+\left (-2 \sin \left (\lambda x +\mu \right )^{k} x^{n} a b -2 a c \sin \left (\lambda x +\mu \right )^{k}\right ) a \sin \left (\lambda x +\mu \right )^{k}\right ) u^{\prime }\left (x \right )+a^{2} \sin \left (\lambda x +\mu \right )^{2 k} \left (\sin \left (\lambda x +\mu \right )^{k} x^{2 n} a \,b^{2}+2 \sin \left (\lambda x +\mu \right )^{k} x^{n} a b c +\sin \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
\[ u \left (x \right ) = {\mathrm e}^{-\frac {\left (\int \left (2 a \left (b \,x^{n}+c \right ) \sin \left (\lambda x +\mu \right )^{k}-\cot \left (\lambda x +\mu \right ) \lambda \left (k -1\right )\right )d x \right )}{2}} \sin \left (\lambda x +\mu \right ) \left (c_{1} \operatorname {LegendreP}\left (\frac {k}{2}-\frac {1}{2}, \frac {1}{2}+\frac {k}{2}, \cos \left (\lambda x +\mu \right )\right )+c_{2} \operatorname {LegendreQ}\left (\frac {k}{2}-\frac {1}{2}, \frac {1}{2}+\frac {k}{2}, \cos \left (\lambda x +\mu \right )\right )\right ) \] The above shows that \[ u^{\prime }\left (x \right ) = -a \sin \left (\lambda x +\mu \right )^{k} {\mathrm e}^{-\frac {\left (\int \left (2 a \left (b \,x^{n}+c \right ) \sin \left (\lambda x +\mu \right )^{k}-\cot \left (\lambda x +\mu \right ) \lambda \left (k -1\right )\right )d x \right )}{2}} \sin \left (\lambda x +\mu \right ) \left (c_{1} \operatorname {LegendreP}\left (\frac {k}{2}-\frac {1}{2}, \frac {1}{2}+\frac {k}{2}, \cos \left (\lambda x +\mu \right )\right )+c_{2} \operatorname {LegendreQ}\left (\frac {k}{2}-\frac {1}{2}, \frac {1}{2}+\frac {k}{2}, \cos \left (\lambda x +\mu \right )\right )\right ) \left (b \,x^{n}+c \right ) \] Using the above in (1) gives the solution \[ y = b \,x^{n}+c \] Dividing both numerator and denominator by \(c_{1}\) gives, after renaming the constant \(\frac {c_{2}}{c_{1}}=c_{3}\) the following solution
\[ y = b \,x^{n}+c \]
The solution(s) found are the following \begin{align*} \tag{1} y &= b \,x^{n}+c \\ \end{align*}
Verification of solutions
\[ y = b \,x^{n}+c \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-a \sin \left (\lambda x +\mu \right )^{k} \left (y-b \,x^{n}-c \right )^{2}=b n \,x^{n -1} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=a \sin \left (\lambda x +\mu \right )^{k} \left (y-b \,x^{n}-c \right )^{2}+b n \,x^{n -1} \end {array} \]
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: 0.0 (sec). Leaf size: 37
dsolve(diff(y(x),x)=a*sin(lambda*x+mu)^k*(y(x)-b*x^n-c)^2+b*n*x^(n-1),y(x), singsol=all)
\[ y \left (x \right ) = b \,x^{n}+c +\frac {1}{c_{1} -a \left (\int \left (\sin \left (x \lambda \right ) \cos \left (\mu \right )+\cos \left (x \lambda \right ) \sin \left (\mu \right )\right )^{k}d x \right )} \]
✓ Solution by Mathematica
Time used: 5.928 (sec). Leaf size: 93
DSolve[y'[x]==a*Sin[\[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 \sqrt {\cos ^2(\mu +\lambda x)} \sec (\mu +\lambda x) \sin ^{k+1}(\mu +\lambda x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {k+1}{2},\frac {k+3}{2},\sin ^2(x \lambda +\mu )\right )}{(k+1) \lambda }+c_1}+b x^n+c \\ y(x)\to b x^n+c \\ \end{align*}