Internal problem ID [10578]
Internal file name [OUTPUT/9526_Monday_June_06_2022_03_04_23_PM_74571615/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: 22.
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]
\[ \boxed {y^{\prime }-\lambda \arctan \left (x \right )^{n} y^{2}-a y=b a -b^{2} \lambda \arctan \left (x \right )^{n}} \]
In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= \lambda \arctan \left (x \right )^{n} y^{2}+a y +b a -b^{2} \lambda \arctan \left (x \right )^{n} \end {align*}
This is a Riccati ODE. Comparing the ODE to solve \[ y' = \lambda \arctan \left (x \right )^{n} y^{2}+a y +b a -b^{2} \lambda \arctan \left (x \right )^{n} \] With Riccati ODE standard form \[ y' = f_0(x)+ f_1(x)y+f_2(x)y^{2} \] Shows that \(f_0(x)=b a -b^{2} \lambda \arctan \left (x \right )^{n}\), \(f_1(x)=a\) and \(f_2(x)=\arctan \left (x \right )^{n} \lambda \). Let \begin {align*} y &= \frac {-u'}{f_2 u} \\ &= \frac {-u'}{\arctan \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 {\arctan \left (x \right )^{n} n \lambda }{\left (x^{2}+1\right ) \arctan \left (x \right )}\\ f_1 f_2 &=a \lambda \arctan \left (x \right )^{n}\\ f_2^2 f_0 &=\arctan \left (x \right )^{2 n} \lambda ^{2} \left (b a -b^{2} \lambda \arctan \left (x \right )^{n}\right ) \end {align*}
Substituting the above terms back in equation (2) gives \begin {align*} \arctan \left (x \right )^{n} \lambda u^{\prime \prime }\left (x \right )-\left (\frac {\arctan \left (x \right )^{n} n \lambda }{\left (x^{2}+1\right ) \arctan \left (x \right )}+a \lambda \arctan \left (x \right )^{n}\right ) u^{\prime }\left (x \right )+\arctan \left (x \right )^{2 n} \lambda ^{2} \left (b a -b^{2} \lambda \arctan \left (x \right )^{n}\right ) u \left (x \right ) &=0 \end {align*}
Solving the above ODE (this ode solved using Maple, not this program), gives
\[ u \left (x \right ) = \operatorname {DESol}\left (\left \{\textit {\_Y}^{\prime \prime }\left (x \right )-\frac {n \textit {\_Y}^{\prime }\left (x \right )}{\left (x^{2}+1\right ) \arctan \left (x \right )}-a \textit {\_Y}^{\prime }\left (x \right )-\arctan \left (x \right )^{2 n} b^{2} \lambda ^{2} \textit {\_Y} \left (x \right )+\arctan \left (x \right )^{n} a b \lambda \textit {\_Y} \left (x \right )\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right ) \] The above shows that \[ u^{\prime }\left (x \right ) = \frac {\partial }{\partial x}\operatorname {DESol}\left (\left \{\textit {\_Y}^{\prime \prime }\left (x \right )-\frac {n \textit {\_Y}^{\prime }\left (x \right )}{\left (x^{2}+1\right ) \arctan \left (x \right )}-a \textit {\_Y}^{\prime }\left (x \right )-\arctan \left (x \right )^{2 n} b^{2} \lambda ^{2} \textit {\_Y} \left (x \right )+\arctan \left (x \right )^{n} a b \lambda \textit {\_Y} \left (x \right )\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right ) \] Using the above in (1) gives the solution \[ y = -\frac {\left (\frac {\partial }{\partial x}\operatorname {DESol}\left (\left \{\textit {\_Y}^{\prime \prime }\left (x \right )-\frac {n \textit {\_Y}^{\prime }\left (x \right )}{\left (x^{2}+1\right ) \arctan \left (x \right )}-a \textit {\_Y}^{\prime }\left (x \right )-\arctan \left (x \right )^{2 n} b^{2} \lambda ^{2} \textit {\_Y} \left (x \right )+\arctan \left (x \right )^{n} a b \lambda \textit {\_Y} \left (x \right )\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )\right ) \arctan \left (x \right )^{-n}}{\lambda \operatorname {DESol}\left (\left \{\textit {\_Y}^{\prime \prime }\left (x \right )-\frac {n \textit {\_Y}^{\prime }\left (x \right )}{\left (x^{2}+1\right ) \arctan \left (x \right )}-a \textit {\_Y}^{\prime }\left (x \right )-\arctan \left (x \right )^{2 n} b^{2} \lambda ^{2} \textit {\_Y} \left (x \right )+\arctan \left (x \right )^{n} a b \lambda \textit {\_Y} \left (x \right )\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )} \] Dividing both numerator and denominator by \(c_{1}\) gives, after renaming the constant \(\frac {c_{2}}{c_{1}}=c_{3}\) the following solution
\[ y = -\frac {\left (\frac {\partial }{\partial x}\operatorname {DESol}\left (\left \{\frac {-b^{2} \textit {\_Y} \left (x \right ) \lambda ^{2} \arctan \left (x \right )^{1+2 n} \left (x^{2}+1\right )+a b \lambda \textit {\_Y} \left (x \right ) \arctan \left (x \right )^{n +1} \left (x^{2}+1\right )+\textit {\_Y}^{\prime \prime }\left (x \right ) \left (x^{2}+1\right ) \arctan \left (x \right )-\left (a \left (x^{2}+1\right ) \arctan \left (x \right )+n \right ) \textit {\_Y}^{\prime }\left (x \right )}{\left (x^{2}+1\right ) \arctan \left (x \right )}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )\right ) \arctan \left (x \right )^{-n}}{\lambda \operatorname {DESol}\left (\left \{\frac {-b^{2} \textit {\_Y} \left (x \right ) \lambda ^{2} \arctan \left (x \right )^{1+2 n} \left (x^{2}+1\right )+a b \lambda \textit {\_Y} \left (x \right ) \arctan \left (x \right )^{n +1} \left (x^{2}+1\right )+\textit {\_Y}^{\prime \prime }\left (x \right ) \left (x^{2}+1\right ) \arctan \left (x \right )-\left (a \left (x^{2}+1\right ) \arctan \left (x \right )+n \right ) \textit {\_Y}^{\prime }\left (x \right )}{\left (x^{2}+1\right ) \arctan \left (x \right )}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )} \]
The solution(s) found are the following \begin{align*} \tag{1} y &= -\frac {\left (\frac {\partial }{\partial x}\operatorname {DESol}\left (\left \{\frac {-b^{2} \textit {\_Y} \left (x \right ) \lambda ^{2} \arctan \left (x \right )^{1+2 n} \left (x^{2}+1\right )+a b \lambda \textit {\_Y} \left (x \right ) \arctan \left (x \right )^{n +1} \left (x^{2}+1\right )+\textit {\_Y}^{\prime \prime }\left (x \right ) \left (x^{2}+1\right ) \arctan \left (x \right )-\left (a \left (x^{2}+1\right ) \arctan \left (x \right )+n \right ) \textit {\_Y}^{\prime }\left (x \right )}{\left (x^{2}+1\right ) \arctan \left (x \right )}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )\right ) \arctan \left (x \right )^{-n}}{\lambda \operatorname {DESol}\left (\left \{\frac {-b^{2} \textit {\_Y} \left (x \right ) \lambda ^{2} \arctan \left (x \right )^{1+2 n} \left (x^{2}+1\right )+a b \lambda \textit {\_Y} \left (x \right ) \arctan \left (x \right )^{n +1} \left (x^{2}+1\right )+\textit {\_Y}^{\prime \prime }\left (x \right ) \left (x^{2}+1\right ) \arctan \left (x \right )-\left (a \left (x^{2}+1\right ) \arctan \left (x \right )+n \right ) \textit {\_Y}^{\prime }\left (x \right )}{\left (x^{2}+1\right ) \arctan \left (x \right )}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )} \\ \end{align*}
Verification of solutions
\[ y = -\frac {\left (\frac {\partial }{\partial x}\operatorname {DESol}\left (\left \{\frac {-b^{2} \textit {\_Y} \left (x \right ) \lambda ^{2} \arctan \left (x \right )^{1+2 n} \left (x^{2}+1\right )+a b \lambda \textit {\_Y} \left (x \right ) \arctan \left (x \right )^{n +1} \left (x^{2}+1\right )+\textit {\_Y}^{\prime \prime }\left (x \right ) \left (x^{2}+1\right ) \arctan \left (x \right )-\left (a \left (x^{2}+1\right ) \arctan \left (x \right )+n \right ) \textit {\_Y}^{\prime }\left (x \right )}{\left (x^{2}+1\right ) \arctan \left (x \right )}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )\right ) \arctan \left (x \right )^{-n}}{\lambda \operatorname {DESol}\left (\left \{\frac {-b^{2} \textit {\_Y} \left (x \right ) \lambda ^{2} \arctan \left (x \right )^{1+2 n} \left (x^{2}+1\right )+a b \lambda \textit {\_Y} \left (x \right ) \arctan \left (x \right )^{n +1} \left (x^{2}+1\right )+\textit {\_Y}^{\prime \prime }\left (x \right ) \left (x^{2}+1\right ) \arctan \left (x \right )-\left (a \left (x^{2}+1\right ) \arctan \left (x \right )+n \right ) \textit {\_Y}^{\prime }\left (x \right )}{\left (x^{2}+1\right ) \arctan \left (x \right )}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-\lambda \arctan \left (x \right )^{n} y^{2}-a y=b a -b^{2} \lambda \arctan \left (x \right )^{n} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\lambda \arctan \left (x \right )^{n} y^{2}+a y+b a -b^{2} \lambda \arctan \left (x \right )^{n} \end {array} \]
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)*a*x^2+a*arctan(x)+n)*(diff(y(x), x))/((x^2+1)*arctan(x))+ar 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)-(arctan(x)^n*lambda*y(x)^2+y(x)+y(x)*a*x+x^2*(a*b-b^2*lambda*arctan(x)^n))/x, y(x 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)] <- symmetry pattern of the form [0, F(x)*G(y)] successful <- Riccati with symmetry pattern of the form [0,F(x)*G(y)] successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 87
dsolve(diff(y(x),x)=lambda*arctan(x)^n*y(x)^2+a*y(x)+a*b-b^2*lambda*arctan(x)^n,y(x), singsol=all)
\[ y \left (x \right ) = \frac {-b \lambda \left (\int \arctan \left (x \right )^{n} {\mathrm e}^{-\left (\int \left (2 \arctan \left (x \right )^{n} \lambda b -a \right )d x \right )}d x \right )-c_{1} b -{\mathrm e}^{-\left (\int \left (2 \arctan \left (x \right )^{n} \lambda b -a \right )d x \right )}}{c_{1} +\lambda \left (\int \arctan \left (x \right )^{n} {\mathrm e}^{-\left (\int \left (2 \arctan \left (x \right )^{n} \lambda b -a \right )d x \right )}d x \right )} \]
✓ Solution by Mathematica
Time used: 10.998 (sec). Leaf size: 240
DSolve[y'[x]==\[Lambda]*ArcTan[x]^n*y[x]^2+a*y[x]+a*b-b^2*\[Lambda]*ArcTan[x]^n,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^x\frac {\exp \left (-\int _1^{K[2]}\left (2 b \lambda \arctan (K[1])^n-a\right )dK[1]\right ) \left (-b \lambda \arctan (K[2])^n+\lambda y(x) \arctan (K[2])^n+a\right )}{a n \lambda (b+y(x))}dK[2]+\int _1^{y(x)}\left (-\int _1^x\left (\frac {\exp \left (-\int _1^{K[2]}\left (2 b \lambda \arctan (K[1])^n-a\right )dK[1]\right ) \arctan (K[2])^n}{a n (b+K[3])}-\frac {\exp \left (-\int _1^{K[2]}\left (2 b \lambda \arctan (K[1])^n-a\right )dK[1]\right ) \left (-b \lambda \arctan (K[2])^n+\lambda K[3] \arctan (K[2])^n+a\right )}{a n \lambda (b+K[3])^2}\right )dK[2]-\frac {\exp \left (-\int _1^x\left (2 b \lambda \arctan (K[1])^n-a\right )dK[1]\right )}{a n \lambda (b+K[3])^2}\right )dK[3]=c_1,y(x)\right ] \]