Internal problem ID [9125]
Internal file name [OUTPUT/8060_Monday_June_06_2022_01_35_00_AM_55737954/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 791.
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.
\[ \boxed {y^{\prime }-\frac {2 x^{2} \cosh \left (\frac {1}{x -1}\right )-2 x \cosh \left (\frac {1}{x -1}\right )-1+y^{2}-2 x^{2} y+x^{4}-x +y^{2} x -2 x^{3} y+x^{5}}{\left (x -1\right ) \cosh \left (\frac {1}{x -1}\right )}=0} \]
In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= \frac {2 x^{2} \cosh \left (\frac {1}{x -1}\right )-2 x \cosh \left (\frac {1}{x -1}\right )-1+y^{2}-2 x^{2} y +x^{4}-x +x \,y^{2}-2 x^{3} y +x^{5}}{\left (x -1\right ) \cosh \left (\frac {1}{x -1}\right )} \end {align*}
This is a Riccati ODE. Comparing the ODE to solve \[ y' = \frac {x^{5}}{\left (x -1\right ) \cosh \left (\frac {1}{x -1}\right )}+\frac {x^{4}}{\left (x -1\right ) \cosh \left (\frac {1}{x -1}\right )}-\frac {2 x^{3} y}{\left (x -1\right ) \cosh \left (\frac {1}{x -1}\right )}+\frac {2 x^{2}}{x -1}-\frac {2 x^{2} y}{\left (x -1\right ) \cosh \left (\frac {1}{x -1}\right )}+\frac {x \,y^{2}}{\left (x -1\right ) \cosh \left (\frac {1}{x -1}\right )}-\frac {2 x}{x -1}+\frac {y^{2}}{\left (x -1\right ) \cosh \left (\frac {1}{x -1}\right )}-\frac {x}{\left (x -1\right ) \cosh \left (\frac {1}{x -1}\right )}-\frac {1}{\left (x -1\right ) \cosh \left (\frac {1}{x -1}\right )} \] With Riccati ODE standard form \[ y' = f_0(x)+ f_1(x)y+f_2(x)y^{2} \] Shows that \(f_0(x)=\frac {x^{5}+x^{4}+2 x^{2} \cosh \left (\frac {1}{x -1}\right )-2 x \cosh \left (\frac {1}{x -1}\right )-x -1}{\left (x -1\right ) \cosh \left (\frac {1}{x -1}\right )}\), \(f_1(x)=\frac {-2 x^{3}-2 x^{2}}{\left (x -1\right ) \cosh \left (\frac {1}{x -1}\right )}\) and \(f_2(x)=\frac {x +1}{\left (x -1\right ) \cosh \left (\frac {1}{x -1}\right )}\). Let \begin {align*} y &= \frac {-u'}{f_2 u} \\ &= \frac {-u'}{\frac {\left (x +1\right ) u}{\left (x -1\right ) \cosh \left (\frac {1}{x -1}\right )}} \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 {1}{\left (x -1\right ) \cosh \left (\frac {1}{x -1}\right )}-\frac {x +1}{\left (x -1\right )^{2} \cosh \left (\frac {1}{x -1}\right )}+\frac {\left (x +1\right ) \sinh \left (\frac {1}{x -1}\right )}{\left (x -1\right )^{3} \cosh \left (\frac {1}{x -1}\right )^{2}}\\ f_1 f_2 &=\frac {\left (-2 x^{3}-2 x^{2}\right ) \left (x +1\right )}{\left (x -1\right )^{2} \cosh \left (\frac {1}{x -1}\right )^{2}}\\ f_2^2 f_0 &=\frac {\left (x +1\right )^{2} \left (x^{5}+x^{4}+2 x^{2} \cosh \left (\frac {1}{x -1}\right )-2 x \cosh \left (\frac {1}{x -1}\right )-x -1\right )}{\left (x -1\right )^{3} \cosh \left (\frac {1}{x -1}\right )^{3}} \end {align*}
Substituting the above terms back in equation (2) gives \begin {align*} \frac {\left (x +1\right ) u^{\prime \prime }\left (x \right )}{\left (x -1\right ) \cosh \left (\frac {1}{x -1}\right )}-\left (\frac {1}{\left (x -1\right ) \cosh \left (\frac {1}{x -1}\right )}-\frac {x +1}{\left (x -1\right )^{2} \cosh \left (\frac {1}{x -1}\right )}+\frac {\left (x +1\right ) \sinh \left (\frac {1}{x -1}\right )}{\left (x -1\right )^{3} \cosh \left (\frac {1}{x -1}\right )^{2}}+\frac {\left (-2 x^{3}-2 x^{2}\right ) \left (x +1\right )}{\left (x -1\right )^{2} \cosh \left (\frac {1}{x -1}\right )^{2}}\right ) u^{\prime }\left (x \right )+\frac {\left (x +1\right )^{2} \left (x^{5}+x^{4}+2 x^{2} \cosh \left (\frac {1}{x -1}\right )-2 x \cosh \left (\frac {1}{x -1}\right )-x -1\right ) u \left (x \right )}{\left (x -1\right )^{3} \cosh \left (\frac {1}{x -1}\right )^{3}} &=0 \end {align*}
Solving the above ODE (this ode solved using Maple, not this program), gives Unable to solve. Terminating.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & -x^{5}+2 x^{3} y-x^{4}+y^{\prime } \cosh \left (\frac {1}{x -1}\right ) x -2 x^{2} \cosh \left (\frac {1}{x -1}\right )-y^{2} x +2 x^{2} y-y^{\prime } \cosh \left (\frac {1}{x -1}\right )+2 x \cosh \left (\frac {1}{x -1}\right )-y^{2}+x +1=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {2 x^{2} \cosh \left (\frac {1}{x -1}\right )-2 x \cosh \left (\frac {1}{x -1}\right )-1+y^{2}-2 x^{2} y+x^{4}-x +y^{2} x -2 x^{3} y+x^{5}}{x \cosh \left (\frac {1}{x -1}\right )-\cosh \left (\frac {1}{x -1}\right )} \end {array} \]
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) = -(2*x^5+2*x^4-2*x^3+2*x*cosh(1/(x-1))-x*sinh(1/(x-1))-2*x^2-2*cosh(1/( 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 <- 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 <- 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/(cosh(1/(x-1))*(x-1))+1/((x-1)*cosh(1/(x-1))))*y(x)^2+y(x)+(-2*x^3/((x-1)*cos 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)] <- symmetry pattern of the form [F(x),G(x)*y+H(x)] successful <- Riccati with symmetry pattern of the form [F(x),G(x)*y+H(x)] successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 161
dsolve(diff(y(x),x) = (2*x^2*cosh(1/(x-1))-2*x*cosh(1/(x-1))-1+y(x)^2-2*x^2*y(x)+x^4-x+x*y(x)^2-2*x^3*y(x)+x^5)/(x-1)/cosh(1/(x-1)),y(x), singsol=all)
\[ y \left (x \right ) = \frac {\left (x^{2} {\mathrm e}^{-4 \left (\int \frac {{\mathrm e}^{\frac {1}{x -1}} \left (x +1\right )}{\left (x -1\right ) \left ({\mathrm e}^{\frac {2}{x -1}}+1\right )}d x \right )+4 c_{1}}-x^{2}+{\mathrm e}^{-4 \left (\int \frac {{\mathrm e}^{\frac {1}{x -1}} \left (x +1\right )}{\left (x -1\right ) \left ({\mathrm e}^{\frac {2}{x -1}}+1\right )}d x \right )+4 c_{1}}+1\right ) {\mathrm e}^{4 \left (\int \frac {{\mathrm e}^{\frac {1}{x -1}} \left (x +1\right )}{\left (x -1\right ) \left ({\mathrm e}^{\frac {2}{x -1}}+1\right )}d x \right )}}{{\mathrm e}^{4 c_{1}}-{\mathrm e}^{4 \left (\int \frac {{\mathrm e}^{\frac {1}{x -1}} \left (x +1\right )}{\left (x -1\right ) \left ({\mathrm e}^{\frac {2}{x -1}}+1\right )}d x \right )}} \]
✓ Solution by Mathematica
Time used: 12.411 (sec). Leaf size: 109
DSolve[y'[x] == (Sech[(-1 + x)^(-1)]*(-1 - x + x^4 + x^5 - 2*x*Cosh[(-1 + x)^(-1)] + 2*x^2*Cosh[(-1 + x)^(-1)] - 2*x^2*y[x] - 2*x^3*y[x] + y[x]^2 + x*y[x]^2))/(-1 + x),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\exp \left (\int _1^x\frac {2 (K[5]+1) \text {sech}\left (\frac {1}{K[5]-1}\right )}{K[5]-1}dK[5]\right )}{-\int _1^x\frac {\exp \left (\int _1^{K[6]}\frac {2 (K[5]+1) \text {sech}\left (\frac {1}{K[5]-1}\right )}{K[5]-1}dK[5]\right ) (K[6]+1) \text {sech}\left (\frac {1}{K[6]-1}\right )}{K[6]-1}dK[6]+c_1}+x^2+1 \\ y(x)\to x^2+1 \\ \end{align*}