Internal problem ID [10583]
Internal file name [OUTPUT/9531_Monday_June_06_2022_03_05_13_PM_92844768/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 1.2.7-3. Equations containing
arctangent.
Problem number: 27.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "riccati"
Maple gives the following as the ode type
[_Riccati]
\[ \boxed {x y^{\prime }-\left (a \,x^{2 m} y^{2}+b \,x^{n} y+c \right ) \arctan \left (x \right )^{m}+n y=0} \]
In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= \frac {x^{2 m} \arctan \left (x \right )^{m} a \,y^{2}+x^{n} \arctan \left (x \right )^{m} b y +\arctan \left (x \right )^{m} c -n y}{x} \end {align*}
This is a Riccati ODE. Comparing the ODE to solve \[ y' = \frac {x^{2 m} \arctan \left (x \right )^{m} a \,y^{2}}{x}+\frac {x^{n} \arctan \left (x \right )^{m} b y}{x}+\frac {\arctan \left (x \right )^{m} c}{x}-\frac {n y}{x} \] With Riccati ODE standard form \[ y' = f_0(x)+ f_1(x)y+f_2(x)y^{2} \] Shows that \(f_0(x)=\frac {\arctan \left (x \right )^{m} c}{x}\), \(f_1(x)=\frac {\arctan \left (x \right )^{m} x^{n} b -n}{x}\) and \(f_2(x)=\frac {x^{2 m} \arctan \left (x \right )^{m} a}{x}\). Let \begin {align*} y &= \frac {-u'}{f_2 u} \\ &= \frac {-u'}{\frac {x^{2 m} \arctan \left (x \right )^{m} a 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 {2 x^{2 m} m \arctan \left (x \right )^{m} a}{x^{2}}+\frac {x^{2 m} \arctan \left (x \right )^{m} m a}{\left (x^{2}+1\right ) \arctan \left (x \right ) x}-\frac {x^{2 m} \arctan \left (x \right )^{m} a}{x^{2}}\\ f_1 f_2 &=\frac {\left (\arctan \left (x \right )^{m} x^{n} b -n \right ) x^{2 m} \arctan \left (x \right )^{m} a}{x^{2}}\\ f_2^2 f_0 &=\frac {x^{4 m} \arctan \left (x \right )^{3 m} a^{2} c}{x^{3}} \end {align*}
Substituting the above terms back in equation (2) gives \begin {align*} \frac {x^{2 m} \arctan \left (x \right )^{m} a u^{\prime \prime }\left (x \right )}{x}-\left (\frac {2 x^{2 m} m \arctan \left (x \right )^{m} a}{x^{2}}+\frac {x^{2 m} \arctan \left (x \right )^{m} m a}{\left (x^{2}+1\right ) \arctan \left (x \right ) x}-\frac {x^{2 m} \arctan \left (x \right )^{m} a}{x^{2}}+\frac {\left (\arctan \left (x \right )^{m} x^{n} b -n \right ) x^{2 m} \arctan \left (x \right )^{m} a}{x^{2}}\right ) u^{\prime }\left (x \right )+\frac {x^{4 m} \arctan \left (x \right )^{3 m} a^{2} c u \left (x \right )}{x^{3}} &=0 \end {align*}
Solving the above ODE (this ode solved using Maple, not this program), gives
\[ u \left (x \right ) = \operatorname {DESol}\left (\left \{\textit {\_Y}^{\prime \prime }\left (x \right )-\frac {2 m \textit {\_Y}^{\prime }\left (x \right )}{x}-\frac {m \textit {\_Y}^{\prime }\left (x \right )}{\left (x^{2}+1\right ) \arctan \left (x \right )}+\frac {\textit {\_Y}^{\prime }\left (x \right )}{x}-b \,x^{n -1} \arctan \left (x \right )^{m} \textit {\_Y}^{\prime }\left (x \right )+\frac {n \textit {\_Y}^{\prime }\left (x \right )}{x}+a c \,x^{2 m -2} \textit {\_Y} \left (x \right ) \arctan \left (x \right )^{2 m}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right ) \] The above shows that \[ u^{\prime }\left (x \right ) = \frac {\partial }{\partial x}\operatorname {DESol}\left (\left \{\textit {\_Y}^{\prime \prime }\left (x \right )-\frac {2 m \textit {\_Y}^{\prime }\left (x \right )}{x}-\frac {m \textit {\_Y}^{\prime }\left (x \right )}{\left (x^{2}+1\right ) \arctan \left (x \right )}+\frac {\textit {\_Y}^{\prime }\left (x \right )}{x}-b \,x^{n -1} \arctan \left (x \right )^{m} \textit {\_Y}^{\prime }\left (x \right )+\frac {n \textit {\_Y}^{\prime }\left (x \right )}{x}+a c \,x^{2 m -2} \textit {\_Y} \left (x \right ) \arctan \left (x \right )^{2 m}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right ) \] Using the above in (1) gives the solution \[ y = -\frac {\left (\frac {\partial }{\partial x}\operatorname {DESol}\left (\left \{\textit {\_Y}^{\prime \prime }\left (x \right )-\frac {2 m \textit {\_Y}^{\prime }\left (x \right )}{x}-\frac {m \textit {\_Y}^{\prime }\left (x \right )}{\left (x^{2}+1\right ) \arctan \left (x \right )}+\frac {\textit {\_Y}^{\prime }\left (x \right )}{x}-b \,x^{n -1} \arctan \left (x \right )^{m} \textit {\_Y}^{\prime }\left (x \right )+\frac {n \textit {\_Y}^{\prime }\left (x \right )}{x}+a c \,x^{2 m -2} \textit {\_Y} \left (x \right ) \arctan \left (x \right )^{2 m}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )\right ) x^{-2 m} \arctan \left (x \right )^{-m} x}{a \operatorname {DESol}\left (\left \{\textit {\_Y}^{\prime \prime }\left (x \right )-\frac {2 m \textit {\_Y}^{\prime }\left (x \right )}{x}-\frac {m \textit {\_Y}^{\prime }\left (x \right )}{\left (x^{2}+1\right ) \arctan \left (x \right )}+\frac {\textit {\_Y}^{\prime }\left (x \right )}{x}-b \,x^{n -1} \arctan \left (x \right )^{m} \textit {\_Y}^{\prime }\left (x \right )+\frac {n \textit {\_Y}^{\prime }\left (x \right )}{x}+a c \,x^{2 m -2} \textit {\_Y} \left (x \right ) \arctan \left (x \right )^{2 m}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )} \] Dividing both numerator and denominator by \(c_{1}\) gives, after renaming the constant \(\frac {c_{2}}{c_{1}}=c_{3}\) the following solution
\[ y = -\frac {x^{-2 m +1} \arctan \left (x \right )^{-m} \left (\frac {\partial }{\partial x}\operatorname {DESol}\left (\left \{\frac {a c \,x^{2 m -1} \textit {\_Y} \left (x \right ) \arctan \left (x \right )^{1+2 m} \left (x^{2}+1\right )-b \,x^{n} \arctan \left (x \right )^{1+m} \left (x^{2}+1\right ) \textit {\_Y}^{\prime }\left (x \right )+\textit {\_Y}^{\prime \prime }\left (x \right ) \arctan \left (x \right ) x \left (x^{2}+1\right )-2 \left (\left (m -\frac {n}{2}-\frac {1}{2}\right ) \left (x^{2}+1\right ) \arctan \left (x \right )+\frac {m x}{2}\right ) \textit {\_Y}^{\prime }\left (x \right )}{\left (x^{2}+1\right ) \arctan \left (x \right ) x}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )\right )}{a \operatorname {DESol}\left (\left \{\frac {a c \textit {\_Y} \left (x \right ) \left (x^{2+2 m}+x^{2 m}\right ) \arctan \left (x \right )^{1+2 m}-\textit {\_Y}^{\prime }\left (x \right ) b \left (x^{n +1}+x^{n +3}\right ) \arctan \left (x \right )^{1+m}-2 \left (-\frac {\textit {\_Y}^{\prime \prime }\left (x \right ) \arctan \left (x \right ) x \left (x^{2}+1\right )}{2}+\left (\left (m -\frac {n}{2}-\frac {1}{2}\right ) \left (x^{2}+1\right ) \arctan \left (x \right )+\frac {m x}{2}\right ) \textit {\_Y}^{\prime }\left (x \right )\right ) x}{x^{2} \left (x^{2}+1\right ) \arctan \left (x \right )}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )} \]
The solution(s) found are the following \begin{align*} \tag{1} y &= -\frac {x^{-2 m +1} \arctan \left (x \right )^{-m} \left (\frac {\partial }{\partial x}\operatorname {DESol}\left (\left \{\frac {a c \,x^{2 m -1} \textit {\_Y} \left (x \right ) \arctan \left (x \right )^{1+2 m} \left (x^{2}+1\right )-b \,x^{n} \arctan \left (x \right )^{1+m} \left (x^{2}+1\right ) \textit {\_Y}^{\prime }\left (x \right )+\textit {\_Y}^{\prime \prime }\left (x \right ) \arctan \left (x \right ) x \left (x^{2}+1\right )-2 \left (\left (m -\frac {n}{2}-\frac {1}{2}\right ) \left (x^{2}+1\right ) \arctan \left (x \right )+\frac {m x}{2}\right ) \textit {\_Y}^{\prime }\left (x \right )}{\left (x^{2}+1\right ) \arctan \left (x \right ) x}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )\right )}{a \operatorname {DESol}\left (\left \{\frac {a c \textit {\_Y} \left (x \right ) \left (x^{2+2 m}+x^{2 m}\right ) \arctan \left (x \right )^{1+2 m}-\textit {\_Y}^{\prime }\left (x \right ) b \left (x^{n +1}+x^{n +3}\right ) \arctan \left (x \right )^{1+m}-2 \left (-\frac {\textit {\_Y}^{\prime \prime }\left (x \right ) \arctan \left (x \right ) x \left (x^{2}+1\right )}{2}+\left (\left (m -\frac {n}{2}-\frac {1}{2}\right ) \left (x^{2}+1\right ) \arctan \left (x \right )+\frac {m x}{2}\right ) \textit {\_Y}^{\prime }\left (x \right )\right ) x}{x^{2} \left (x^{2}+1\right ) \arctan \left (x \right )}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )} \\ \end{align*}
Verification of solutions
\[ y = -\frac {x^{-2 m +1} \arctan \left (x \right )^{-m} \left (\frac {\partial }{\partial x}\operatorname {DESol}\left (\left \{\frac {a c \,x^{2 m -1} \textit {\_Y} \left (x \right ) \arctan \left (x \right )^{1+2 m} \left (x^{2}+1\right )-b \,x^{n} \arctan \left (x \right )^{1+m} \left (x^{2}+1\right ) \textit {\_Y}^{\prime }\left (x \right )+\textit {\_Y}^{\prime \prime }\left (x \right ) \arctan \left (x \right ) x \left (x^{2}+1\right )-2 \left (\left (m -\frac {n}{2}-\frac {1}{2}\right ) \left (x^{2}+1\right ) \arctan \left (x \right )+\frac {m x}{2}\right ) \textit {\_Y}^{\prime }\left (x \right )}{\left (x^{2}+1\right ) \arctan \left (x \right ) x}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )\right )}{a \operatorname {DESol}\left (\left \{\frac {a c \textit {\_Y} \left (x \right ) \left (x^{2+2 m}+x^{2 m}\right ) \arctan \left (x \right )^{1+2 m}-\textit {\_Y}^{\prime }\left (x \right ) b \left (x^{n +1}+x^{n +3}\right ) \arctan \left (x \right )^{1+m}-2 \left (-\frac {\textit {\_Y}^{\prime \prime }\left (x \right ) \arctan \left (x \right ) x \left (x^{2}+1\right )}{2}+\left (\left (m -\frac {n}{2}-\frac {1}{2}\right ) \left (x^{2}+1\right ) \arctan \left (x \right )+\frac {m x}{2}\right ) \textit {\_Y}^{\prime }\left (x \right )\right ) x}{x^{2} \left (x^{2}+1\right ) \arctan \left (x \right )}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x y^{\prime }-\left (a \,x^{2 m} y^{2}+b \,x^{n} y+c \right ) \arctan \left (x \right )^{m}+n y=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {\left (a \,x^{2 m} y^{2}+b \,x^{n} y+c \right ) \arctan \left (x \right )^{m}-n y}{x} \end {array} \]
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 differential order: 1; looking for linear symmetries trying exact Looking for potential symmetries trying Riccati trying Riccati sub-methods: trying Riccati_symmetries trying Riccati to 2nd Order -> Calling odsolve with the ODE`, diff(diff(y(x), x), x) = (arctan(x)^m*arctan(x)*x^(n-1)*b*x^3+b*x^(n-1)*arctan(x)^m*arctan(x)*x Methods for second order ODEs: --- Trying classification methods --- trying a symmetry of the form [xi=0, eta=F(x)] checking if the LODE is missing y -> Heun: Equivalence to the GHE or one of its 4 confluent cases under a power @ Moebius -> trying a solution of the form r0(x) * Y + r1(x) * Y where Y = exp(int(r(x), dx)) * 2F1([a1, a2], [b1], f) -> Trying changes of variables to rationalize or make the ODE simpler trying a symmetry of the form [xi=0, eta=F(x)] checking if the LODE is missing y -> Heun: Equivalence to the GHE or one of its 4 confluent cases under a power @ Moebius -> trying a solution of the form r0(x) * Y + r1(x) * Y where Y = exp(int(r(x), dx)) * 2F1([a1, a2], [b1], f) trying a symmetry of the form [xi=0, eta=F(x)] trying 2nd order exact linear trying symmetries linear in x and y(x) trying to convert to a linear ODE with constant coefficients -> trying with_periodic_functions in the coefficients <- unable to find a useful change of variables trying a symmetry of the form [xi=0, eta=F(x)] trying 2nd order exact linear trying symmetries linear in x and y(x) trying to convert to a linear ODE with constant coefficients trying 2nd order, integrating factor of the form mu(x,y) trying a symmetry of the form [xi=0, eta=F(x)] checking if the LODE is missing y -> Heun: Equivalence to the GHE or one of its 4 confluent cases under a power @ Moebius -> trying a solution of the form r0(x) * Y + r1(x) * Y where Y = exp(int(r(x), dx)) * 2F1([a1, a2], [b1], f) -> Trying changes of variables to rationalize or make the ODE simpler trying a symmetry of the form [xi=0, eta=F(x)] checking if the LODE is missing y -> Heun: Equivalence to the GHE or one of its 4 confluent cases under a power @ Moebius -> trying a solution of the form r0(x) * Y + r1(x) * Y where Y = exp(int(r(x), dx)) * 2F1([a1, a2], [b1], f) trying a symmetry of the form [xi=0, eta=F(x)] trying 2nd order exact linear trying symmetries linear in x and y(x) trying to convert to a linear ODE with constant coefficients -> trying with_periodic_functions in the coefficients <- unable to find a useful change of variables trying a symmetry of the form [xi=0, eta=F(x)] trying to convert to an ODE of Bessel type -> Trying a change of variables to reduce to Bernoulli -> Calling odsolve with the ODE`, diff(y(x), x)-(x^(-1+2*m)*a*arctan(x)^m*y(x)^2+y(x)+(b*x^(n-1)*arctan(x)^m-n/x)*y(x)*x+x*arctan 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: trying Riccati_symmetries trying inverse_Riccati trying 1st order ODE linearizable_by_differentiation -> trying a symmetry pattern of the form [F(x)*G(y), 0] -> trying a symmetry pattern of the form [0, F(x)*G(y)] -> trying a symmetry pattern of the form [F(x),G(x)*y+H(x)] trying inverse_Riccati trying 1st order ODE linearizable_by_differentiation --- Trying Lie symmetry methods, 1st order --- `, `-> Computing symmetries using: way = 4 `, `-> Computing symmetries using: way = 2 `, `-> Computing symmetries using: way = 6`
✗ Solution by Maple
dsolve(x*diff(y(x),x)=(a*x^(2*m)*y(x)^2+b*x^n*y(x)+c)*arctan(x)^m-n*y(x),y(x), singsol=all)
\[ \text {No solution found} \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[x*y'[x]==(a*x^(2*m)*y[x]^2+b*x^n*y[x]+c)*ArcTan[x]^m-n*y[x],y[x],x,IncludeSingularSolutions -> True]
Not solved