Internal problem ID [10572]
Internal file name [OUTPUT/9520_Monday_June_06_2022_03_03_27_PM_85306196/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 1.2.7-2. Equations containing
arccosine.
Problem number: 16.
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]
\[ \boxed {y^{\prime }-\lambda \arccos \left (x \right )^{n} \left (y-a \,x^{m}-b \right )^{2}=a m \,x^{m -1}} \]
In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= \arccos \left (x \right )^{n} x^{2 m} a^{2} \lambda +2 \arccos \left (x \right )^{n} x^{m} a b \lambda -2 \arccos \left (x \right )^{n} x^{m} a \lambda y +b^{2} \lambda \arccos \left (x \right )^{n}-2 \arccos \left (x \right )^{n} b \lambda y +\arccos \left (x \right )^{n} \lambda \,y^{2}+a m \,x^{m -1} \end {align*}
This is a Riccati ODE. Comparing the ODE to solve \[ y' = \arccos \left (x \right )^{n} x^{2 m} a^{2} \lambda +2 \arccos \left (x \right )^{n} x^{m} a b \lambda -2 \arccos \left (x \right )^{n} x^{m} a \lambda y +b^{2} \lambda \arccos \left (x \right )^{n}-2 \arccos \left (x \right )^{n} b \lambda y +\arccos \left (x \right )^{n} \lambda \,y^{2}+\frac {a m \,x^{m}}{x} \] With Riccati ODE standard form \[ y' = f_0(x)+ f_1(x)y+f_2(x)y^{2} \] Shows that \(f_0(x)=\arccos \left (x \right )^{n} x^{2 m} a^{2} \lambda +2 \arccos \left (x \right )^{n} x^{m} a b \lambda +b^{2} \lambda \arccos \left (x \right )^{n}+a m \,x^{m -1}\), \(f_1(x)=-2 \arccos \left (x \right )^{n} x^{m} a \lambda -2 \arccos \left (x \right )^{n} \lambda b\) and \(f_2(x)=\arccos \left (x \right )^{n} \lambda \). Let \begin {align*} y &= \frac {-u'}{f_2 u} \\ &= \frac {-u'}{\arccos \left (x \right )^{n} \lambda 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 {\arccos \left (x \right )^{n} n \lambda }{\sqrt {-x^{2}+1}\, \arccos \left (x \right )}\\ f_1 f_2 &=\left (-2 \arccos \left (x \right )^{n} x^{m} a \lambda -2 \arccos \left (x \right )^{n} \lambda b \right ) \arccos \left (x \right )^{n} \lambda \\ f_2^2 f_0 &=\arccos \left (x \right )^{2 n} \lambda ^{2} \left (\arccos \left (x \right )^{n} x^{2 m} a^{2} \lambda +2 \arccos \left (x \right )^{n} x^{m} a b \lambda +b^{2} \lambda \arccos \left (x \right )^{n}+a m \,x^{m -1}\right ) \end {align*}
Substituting the above terms back in equation (2) gives \begin {align*} \arccos \left (x \right )^{n} \lambda u^{\prime \prime }\left (x \right )-\left (-\frac {\arccos \left (x \right )^{n} n \lambda }{\sqrt {-x^{2}+1}\, \arccos \left (x \right )}+\left (-2 \arccos \left (x \right )^{n} x^{m} a \lambda -2 \arccos \left (x \right )^{n} \lambda b \right ) \arccos \left (x \right )^{n} \lambda \right ) u^{\prime }\left (x \right )+\arccos \left (x \right )^{2 n} \lambda ^{2} \left (\arccos \left (x \right )^{n} x^{2 m} a^{2} \lambda +2 \arccos \left (x \right )^{n} x^{m} a b \lambda +b^{2} \lambda \arccos \left (x \right )^{n}+a m \,x^{m -1}\right ) u \left (x \right ) &=0 \end {align*}
Solving the above ODE (this ode solved using Maple, not this program), gives
\[ u \left (x \right ) = \operatorname {DESol}\left (\left \{\frac {\frac {n \textit {\_Y}^{\prime }\left (x \right )}{\sqrt {-x^{2}+1}}+\left (\lambda ^{2} \textit {\_Y} \left (x \right ) \left (a^{2} x^{2 m}+2 a b \,x^{m}+b^{2}\right ) \arccos \left (x \right )^{2 n}+\textit {\_Y}^{\prime \prime }\left (x \right )+x^{m -1} \arccos \left (x \right )^{n} a m \lambda \textit {\_Y} \left (x \right )+2 \lambda \left (a \,x^{m}+b \right ) \arccos \left (x \right )^{n} \textit {\_Y}^{\prime }\left (x \right )\right ) \arccos \left (x \right )}{\arccos \left (x \right )}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right ) \] The above shows that \[ u^{\prime }\left (x \right ) = \frac {\partial }{\partial x}\operatorname {DESol}\left (\left \{\frac {\frac {n \textit {\_Y}^{\prime }\left (x \right )}{\sqrt {-x^{2}+1}}+\lambda ^{2} \textit {\_Y} \left (x \right ) \left (a^{2} x^{2 m}+2 a b \,x^{m}+b^{2}\right ) \arccos \left (x \right )^{1+2 n}+\left (a \,x^{m -1} m \textit {\_Y} \left (x \right )+2 \textit {\_Y}^{\prime }\left (x \right ) \left (a \,x^{m}+b \right )\right ) \lambda \arccos \left (x \right )^{n +1}+\textit {\_Y}^{\prime \prime }\left (x \right ) \arccos \left (x \right )}{\arccos \left (x \right )}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right ) \] Using the above in (1) gives the solution \[ y = -\frac {\left (\frac {\partial }{\partial x}\operatorname {DESol}\left (\left \{\frac {\frac {n \textit {\_Y}^{\prime }\left (x \right )}{\sqrt {-x^{2}+1}}+\lambda ^{2} \textit {\_Y} \left (x \right ) \left (a^{2} x^{2 m}+2 a b \,x^{m}+b^{2}\right ) \arccos \left (x \right )^{1+2 n}+\left (a \,x^{m -1} m \textit {\_Y} \left (x \right )+2 \textit {\_Y}^{\prime }\left (x \right ) \left (a \,x^{m}+b \right )\right ) \lambda \arccos \left (x \right )^{n +1}+\textit {\_Y}^{\prime \prime }\left (x \right ) \arccos \left (x \right )}{\arccos \left (x \right )}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )\right ) \arccos \left (x \right )^{-n}}{\lambda \operatorname {DESol}\left (\left \{\frac {\frac {n \textit {\_Y}^{\prime }\left (x \right )}{\sqrt {-x^{2}+1}}+\left (\lambda ^{2} \textit {\_Y} \left (x \right ) \left (a^{2} x^{2 m}+2 a b \,x^{m}+b^{2}\right ) \arccos \left (x \right )^{2 n}+\textit {\_Y}^{\prime \prime }\left (x \right )+x^{m -1} \arccos \left (x \right )^{n} a m \lambda \textit {\_Y} \left (x \right )+2 \lambda \left (a \,x^{m}+b \right ) \arccos \left (x \right )^{n} \textit {\_Y}^{\prime }\left (x \right )\right ) \arccos \left (x \right )}{\arccos \left (x \right )}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )} \] Dividing both numerator and denominator by \(c_{1}\) gives, after renaming the constant \(\frac {c_{2}}{c_{1}}=c_{3}\) the following solution
\[ y = -\frac {\left (\frac {\partial }{\partial x}\operatorname {DESol}\left (\left \{\frac {\left (\lambda ^{2} \textit {\_Y} \left (x \right ) \left (a^{2} x^{2 m}+2 a b \,x^{m}+b^{2}\right ) \arccos \left (x \right )^{1+2 n}+\left (a \,x^{m -1} m \textit {\_Y} \left (x \right )+2 \textit {\_Y}^{\prime }\left (x \right ) \left (a \,x^{m}+b \right )\right ) \lambda \arccos \left (x \right )^{n +1}+\textit {\_Y}^{\prime \prime }\left (x \right ) \arccos \left (x \right )\right ) \sqrt {-x^{2}+1}+n \textit {\_Y}^{\prime }\left (x \right )}{\sqrt {-x^{2}+1}\, \arccos \left (x \right )}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )\right ) \arccos \left (x \right )^{-n}}{\lambda \operatorname {DESol}\left (\left \{\frac {\left (\lambda ^{2} \textit {\_Y} \left (x \right ) \left (a^{2} x^{2 m}+2 a b \,x^{m}+b^{2}\right ) \arccos \left (x \right )^{1+2 n}+\left (a \,x^{m -1} m \textit {\_Y} \left (x \right )+2 \textit {\_Y}^{\prime }\left (x \right ) \left (a \,x^{m}+b \right )\right ) \lambda \arccos \left (x \right )^{n +1}+\textit {\_Y}^{\prime \prime }\left (x \right ) \arccos \left (x \right )\right ) \sqrt {-x^{2}+1}+n \textit {\_Y}^{\prime }\left (x \right )}{\sqrt {-x^{2}+1}\, \arccos \left (x \right )}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )} \]
The solution(s) found are the following \begin{align*} \tag{1} y &= -\frac {\left (\frac {\partial }{\partial x}\operatorname {DESol}\left (\left \{\frac {\left (\lambda ^{2} \textit {\_Y} \left (x \right ) \left (a^{2} x^{2 m}+2 a b \,x^{m}+b^{2}\right ) \arccos \left (x \right )^{1+2 n}+\left (a \,x^{m -1} m \textit {\_Y} \left (x \right )+2 \textit {\_Y}^{\prime }\left (x \right ) \left (a \,x^{m}+b \right )\right ) \lambda \arccos \left (x \right )^{n +1}+\textit {\_Y}^{\prime \prime }\left (x \right ) \arccos \left (x \right )\right ) \sqrt {-x^{2}+1}+n \textit {\_Y}^{\prime }\left (x \right )}{\sqrt {-x^{2}+1}\, \arccos \left (x \right )}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )\right ) \arccos \left (x \right )^{-n}}{\lambda \operatorname {DESol}\left (\left \{\frac {\left (\lambda ^{2} \textit {\_Y} \left (x \right ) \left (a^{2} x^{2 m}+2 a b \,x^{m}+b^{2}\right ) \arccos \left (x \right )^{1+2 n}+\left (a \,x^{m -1} m \textit {\_Y} \left (x \right )+2 \textit {\_Y}^{\prime }\left (x \right ) \left (a \,x^{m}+b \right )\right ) \lambda \arccos \left (x \right )^{n +1}+\textit {\_Y}^{\prime \prime }\left (x \right ) \arccos \left (x \right )\right ) \sqrt {-x^{2}+1}+n \textit {\_Y}^{\prime }\left (x \right )}{\sqrt {-x^{2}+1}\, \arccos \left (x \right )}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )} \\ \end{align*}
Verification of solutions
\[ y = -\frac {\left (\frac {\partial }{\partial x}\operatorname {DESol}\left (\left \{\frac {\left (\lambda ^{2} \textit {\_Y} \left (x \right ) \left (a^{2} x^{2 m}+2 a b \,x^{m}+b^{2}\right ) \arccos \left (x \right )^{1+2 n}+\left (a \,x^{m -1} m \textit {\_Y} \left (x \right )+2 \textit {\_Y}^{\prime }\left (x \right ) \left (a \,x^{m}+b \right )\right ) \lambda \arccos \left (x \right )^{n +1}+\textit {\_Y}^{\prime \prime }\left (x \right ) \arccos \left (x \right )\right ) \sqrt {-x^{2}+1}+n \textit {\_Y}^{\prime }\left (x \right )}{\sqrt {-x^{2}+1}\, \arccos \left (x \right )}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )\right ) \arccos \left (x \right )^{-n}}{\lambda \operatorname {DESol}\left (\left \{\frac {\left (\lambda ^{2} \textit {\_Y} \left (x \right ) \left (a^{2} x^{2 m}+2 a b \,x^{m}+b^{2}\right ) \arccos \left (x \right )^{1+2 n}+\left (a \,x^{m -1} m \textit {\_Y} \left (x \right )+2 \textit {\_Y}^{\prime }\left (x \right ) \left (a \,x^{m}+b \right )\right ) \lambda \arccos \left (x \right )^{n +1}+\textit {\_Y}^{\prime \prime }\left (x \right ) \arccos \left (x \right )\right ) \sqrt {-x^{2}+1}+n \textit {\_Y}^{\prime }\left (x \right )}{\sqrt {-x^{2}+1}\, \arccos \left (x \right )}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-\lambda \arccos \left (x \right )^{n} \left (y-a \,x^{m}-b \right )^{2}=a m \,x^{m -1} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\lambda \arccos \left (x \right )^{n} \left (y-a \,x^{m}-b \right )^{2}+a m \,x^{m -1} \end {array} \]
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: 165
dsolve(diff(y(x),x)=lambda*arccos(x)^n*(y(x)-a*x^m-b)^2+a*m*x^(m-1),y(x), singsol=all)
\[ y \left (x \right ) = \frac {\lambda \left (a \,x^{m}+b \right ) \left (\left (n +2\right ) \operatorname {LommelS1}\left (n +\frac {1}{2}, \frac {1}{2}, \arccos \left (x \right )\right )-\operatorname {LommelS1}\left (n +\frac {3}{2}, \frac {3}{2}, \arccos \left (x \right )\right ) \arccos \left (x \right )+\arccos \left (x \right )^{n +\frac {3}{2}}\right ) \sqrt {-x^{2}+1}-\left (\lambda \arccos \left (x \right ) \left (a \,x^{1+m}+b x \right ) \operatorname {LommelS1}\left (n +\frac {1}{2}, \frac {1}{2}, \arccos \left (x \right )\right )-\sqrt {\arccos \left (x \right )}\, \left (x^{m} c_{1} a +c_{1} b +1\right )\right ) \left (n +2\right )}{\lambda \left (\left (n +2\right ) \operatorname {LommelS1}\left (n +\frac {1}{2}, \frac {1}{2}, \arccos \left (x \right )\right )-\operatorname {LommelS1}\left (n +\frac {3}{2}, \frac {3}{2}, \arccos \left (x \right )\right ) \arccos \left (x \right )+\arccos \left (x \right )^{n +\frac {3}{2}}\right ) \sqrt {-x^{2}+1}+\left (n +2\right ) \left (-x \lambda \operatorname {LommelS1}\left (n +\frac {1}{2}, \frac {1}{2}, \arccos \left (x \right )\right ) \arccos \left (x \right )+c_{1} \sqrt {\arccos \left (x \right )}\right )} \]
✓ Solution by Mathematica
Time used: 4.776 (sec). Leaf size: 86
DSolve[y'[x]==\[Lambda]*ArcCos[x]^n*(y[x]-a*x^m-b)^2+a*m*x^(m-1),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to a x^m+\frac {1}{-\frac {1}{2} \lambda \arccos (x)^n (-i \arccos (x))^{-n} \Gamma (n+1,-i \arccos (x))-\frac {1}{2} \lambda (i \arccos (x))^{-n} \arccos (x)^n \Gamma (n+1,i \arccos (x))+c_1}+b \\ y(x)\to a x^m+b \\ \end{align*}