Internal problem ID [9323]
Internal file name [OUTPUT/8259_Monday_June_06_2022_02_34_03_AM_45011922/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 990.
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]
\[ \boxed {y^{\prime }+F \left (x \right ) \left (-y^{2}+2 y x^{2}+1-x^{4}\right )=2 x} \]
In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= F \left (x \right ) x^{4}-2 F \left (x \right ) x^{2} y +F \left (x \right ) y^{2}-F \left (x \right )+2 x \end {align*}
This is a Riccati ODE. Comparing the ODE to solve \[ y' = F \left (x \right ) x^{4}-2 F \left (x \right ) x^{2} y +F \left (x \right ) y^{2}-F \left (x \right )+2 x \] With Riccati ODE standard form \[ y' = f_0(x)+ f_1(x)y+f_2(x)y^{2} \] Shows that \(f_0(x)=F \left (x \right ) x^{4}-F \left (x \right )+2 x\), \(f_1(x)=-2 x^{2} F \left (x \right )\) and \(f_2(x)=F \left (x \right )\). Let \begin {align*} y &= \frac {-u'}{f_2 u} \\ &= \frac {-u'}{F \left (x \right ) 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' &=F^{\prime }\left (x \right )\\ f_1 f_2 &=-2 F \left (x \right )^{2} x^{2}\\ f_2^2 f_0 &=F \left (x \right )^{2} \left (F \left (x \right ) x^{4}-F \left (x \right )+2 x \right ) \end {align*}
Substituting the above terms back in equation (2) gives \begin {align*} F \left (x \right ) u^{\prime \prime }\left (x \right )-\left (F^{\prime }\left (x \right )-2 F \left (x \right )^{2} x^{2}\right ) u^{\prime }\left (x \right )+F \left (x \right )^{2} \left (F \left (x \right ) x^{4}-F \left (x \right )+2 x \right ) u \left (x \right ) &=0 \end {align*}
Solving the above ODE (this ode solved using Maple, not this program), gives
\[ u \left (x \right ) = \left ({\mathrm e}^{-\left (\int x^{2} F \left (x \right )d x \right )+\int F \left (x \right )d x}-c_{1} {\mathrm e}^{-\left (\int x^{2} F \left (x \right )d x \right )-\left (\int F \left (x \right )d x \right )}\right ) c_{2} \] The above shows that \[ u^{\prime }\left (x \right ) = c_{2} \left (c_{1} \left (x^{2}+1\right ) {\mathrm e}^{-\left (\int x^{2} F \left (x \right )d x \right )-\left (\int F \left (x \right )d x \right )}+\left (-x^{2}+1\right ) {\mathrm e}^{-\left (\int x^{2} F \left (x \right )d x \right )+\int F \left (x \right )d x}\right ) F \left (x \right ) \] Using the above in (1) gives the solution \[ y = -\frac {c_{1} \left (x^{2}+1\right ) {\mathrm e}^{-\left (\int x^{2} F \left (x \right )d x \right )-\left (\int F \left (x \right )d x \right )}+\left (-x^{2}+1\right ) {\mathrm e}^{-\left (\int x^{2} F \left (x \right )d x \right )+\int F \left (x \right )d x}}{{\mathrm e}^{-\left (\int x^{2} F \left (x \right )d x \right )+\int F \left (x \right )d x}-c_{1} {\mathrm e}^{-\left (\int x^{2} F \left (x \right )d x \right )-\left (\int F \left (x \right )d x \right )}} \] Dividing both numerator and denominator by \(c_{1}\) gives, after renaming the constant \(\frac {c_{2}}{c_{1}}=c_{3}\) the following solution
\[ y = \frac {-{\mathrm e}^{-\left (\int x^{2} F \left (x \right )d x \right )-\left (\int F \left (x \right )d x \right )} c_{3} x^{2}+{\mathrm e}^{-\left (\int x^{2} F \left (x \right )d x \right )+\int F \left (x \right )d x} x^{2}-c_{3} {\mathrm e}^{-\left (\int x^{2} F \left (x \right )d x \right )-\left (\int F \left (x \right )d x \right )}-{\mathrm e}^{-\left (\int x^{2} F \left (x \right )d x \right )+\int F \left (x \right )d x}}{{\mathrm e}^{-\left (\int x^{2} F \left (x \right )d x \right )+\int F \left (x \right )d x}-c_{3} {\mathrm e}^{-\left (\int x^{2} F \left (x \right )d x \right )-\left (\int F \left (x \right )d x \right )}} \]
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {-{\mathrm e}^{-\left (\int x^{2} F \left (x \right )d x \right )-\left (\int F \left (x \right )d x \right )} c_{3} x^{2}+{\mathrm e}^{-\left (\int x^{2} F \left (x \right )d x \right )+\int F \left (x \right )d x} x^{2}-c_{3} {\mathrm e}^{-\left (\int x^{2} F \left (x \right )d x \right )-\left (\int F \left (x \right )d x \right )}-{\mathrm e}^{-\left (\int x^{2} F \left (x \right )d x \right )+\int F \left (x \right )d x}}{{\mathrm e}^{-\left (\int x^{2} F \left (x \right )d x \right )+\int F \left (x \right )d x}-c_{3} {\mathrm e}^{-\left (\int x^{2} F \left (x \right )d x \right )-\left (\int F \left (x \right )d x \right )}} \\ \end{align*}
Verification of solutions
\[ y = \frac {-{\mathrm e}^{-\left (\int x^{2} F \left (x \right )d x \right )-\left (\int F \left (x \right )d x \right )} c_{3} x^{2}+{\mathrm e}^{-\left (\int x^{2} F \left (x \right )d x \right )+\int F \left (x \right )d x} x^{2}-c_{3} {\mathrm e}^{-\left (\int x^{2} F \left (x \right )d x \right )-\left (\int F \left (x \right )d x \right )}-{\mathrm e}^{-\left (\int x^{2} F \left (x \right )d x \right )+\int F \left (x \right )d x}}{{\mathrm e}^{-\left (\int x^{2} F \left (x \right )d x \right )+\int F \left (x \right )d x}-c_{3} {\mathrm e}^{-\left (\int x^{2} F \left (x \right )d x \right )-\left (\int F \left (x \right )d x \right )}} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }+F \left (x \right ) \left (-y^{2}+2 y x^{2}+1-x^{4}\right )=2 x \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-F \left (x \right ) \left (-y^{2}+2 y x^{2}+1-x^{4}\right )+2 x \end {array} \]
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) = -(2*F(x)^2*x^2-(diff(F(x), x)))*(diff(y(x), x))/F(x)-F(x)*(F(x)*x^4-F( 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 <- 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 <- 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)-(F(x)*y(x)^2+y(x)-2*F(x)*x^3*y(x)+x^2*(F(x)*x^4-F(x)+2*x))/x, y(x), explicit` 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: 44
dsolve(diff(y(x),x) = -F(x)*(-y(x)^2+2*x^2*y(x)+1-x^4)+2*x,y(x), singsol=all)
\[ y \left (x \right ) = \frac {-x^{2} {\mathrm e}^{2 \left (\int F \left (x \right )d x \right )}+c_{1} x^{2}+{\mathrm e}^{2 \left (\int F \left (x \right )d x \right )}+c_{1}}{-{\mathrm e}^{2 \left (\int F \left (x \right )d x \right )}+c_{1}} \]
✓ Solution by Mathematica
Time used: 0.297 (sec). Leaf size: 67
DSolve[y'[x] == 2*x - F[x]*(1 - x^4 + 2*x^2*y[x] - y[x]^2),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\exp \left (\int _1^x2 F(K[5])dK[5]\right )}{-\int _1^x\exp \left (\int _1^{K[6]}2 F(K[5])dK[5]\right ) F(K[6])dK[6]+c_1}+x^2+1 \\ y(x)\to x^2+1 \\ \end{align*}