Internal problem ID [10138]
Internal file name [OUTPUT/9085_Monday_June_06_2022_06_27_44_AM_8523807/index.tex
]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1817 (book 6.226).
ODE order: 2.
ODE degree: 1.
The type(s) of ODE detected by this program : "second_order_integrable_as_is", "exact nonlinear second order ode"
Maple gives the following as the ode type
[[_2nd_order, _exact, _nonlinear], [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_poly_yn]]
Unable to solve or complete the solution.
\[ \boxed {y^{\prime } y^{\prime \prime }-x^{2} y y^{\prime }-x y^{2}=0} \]
Integrating both sides of the ODE w.r.t \(x\) gives \begin {align*} \int \left (y^{\prime } y^{\prime \prime }-x^{2} y y^{\prime }-x y^{2}\right )d x &= 0 \\ -\frac {x^{2} y^{2}}{2}+\frac {{y^{\prime }}^{2}}{2} = c_{1} \end {align*}
Which is now solved for \(y\). Solving the given ode for \(y^{\prime }\) results in \(2\) differential equations to solve. Each one of these will generate a solution. The equations generated are \begin {align*} y^{\prime }&=\sqrt {x^{2} y^{2}+2 c_{1}} \tag {1} \\ y^{\prime }&=-\sqrt {x^{2} y^{2}+2 c_{1}} \tag {2} \end {align*}
Now each one of the above ODE is solved.
Solving equation (1)
Unable to determine ODE type.
Unable to determine ODE type.
Solving equation (2)
Unable to determine ODE type.
Unable to determine ODE type.
Writing the ode as \[ y^{\prime } y^{\prime \prime }-x^{2} y y^{\prime }-x y^{2} = 0 \] Integrating both sides of the ODE w.r.t \(x\) gives \begin {align*} \int \left (y^{\prime } y^{\prime \prime }-x^{2} y y^{\prime }-x y^{2}\right )d x &= 0 \\ -\frac {x^{2} y^{2}}{2}+\frac {{y^{\prime }}^{2}}{2} = c_{1} \end {align*}
Which is now solved for \(y\). Solving the given ode for \(y^{\prime }\) results in \(2\) differential equations to solve. Each one of these will generate a solution. The equations generated are \begin {align*} y^{\prime }&=\sqrt {x^{2} y^{2}+2 c_{1}} \tag {1} \\ y^{\prime }&=-\sqrt {x^{2} y^{2}+2 c_{1}} \tag {2} \end {align*}
Now each one of the above ODE is solved.
Solving equation (1)
Unable to determine ODE type.
Unable to determine ODE type.
Solving equation (2)
Unable to determine ODE type.
Unable to determine ODE type.
An exact non-linear second order ode has the form \begin {align*} a_{2} \left (x , y, y^{\prime }\right ) y^{\prime \prime }+a_{1} \left (x , y, y^{\prime }\right ) y^{\prime }+a_{0} \left (x , y, y^{\prime }\right )&=0 \end {align*}
Where the following conditions are satisfied \begin {align*} \frac {\partial a_2}{\partial y} &= \frac {\partial a_1}{\partial y'}\\ \frac {\partial a_2}{\partial x} &= \frac {\partial a_0}{\partial y'}\\ \frac {\partial a_1}{\partial x} &= \frac {\partial a_0}{\partial y} \end {align*}
Looking at the the ode given we see that \begin {align*} a_2 &= y^{\prime }\\ a_1 &= -x^{2} y\\ a_0 &= -x y^{2} \end {align*}
Applying the conditions to the above shows this is a nonlinear exact second order ode. Therefore it can be reduced to first order ode given by \begin {align*} \int {a_2\,d y'} + \int {a_1\,d y} + \int {a_0\,d x} &= c_{1}\\ \int {y^{\prime }\,d y'} + \int {-x^{2} y\,d y} + \int {-x y^{2}\,d x} &= c_{1} \end {align*}
Which results in \begin {align*} \frac {{y^{\prime }}^{2}}{2}-x^{2} y^{2} = c_{1} \end {align*}
Which is now solved Solving the given ode for \(y^{\prime }\) results in \(2\) differential equations to solve. Each one of these will generate a solution. The equations generated are \begin {align*} y^{\prime }&=\sqrt {2 x^{2} y^{2}+2 c_{1}} \tag {1} \\ y^{\prime }&=-\sqrt {2 x^{2} y^{2}+2 c_{1}} \tag {2} \end {align*}
Now each one of the above ODE is solved.
Solving equation (1)
Unable to determine ODE type.
Unable to determine ODE type.
Solving equation (2)
Unable to determine ODE type.
Unable to determine ODE type.
Unable to solve the ode, Terminating
Maple trace
`Methods for second order ODEs: --- Trying classification methods --- trying 2nd order Liouville trying 2nd order WeierstrassP trying 2nd order JacobiSN differential order: 2; trying a linearization to 3rd order trying 2nd order ODE linearizable_by_differentiation trying 2nd order, 2 integrating factors of the form mu(x,y) trying differential order: 2; missing variables -> trying 2nd order, dynamical_symmetries, fully reducible to Abel through one integrating factor of the form G(x,y)/(1+H(x,y)*y)^2 trying 2nd order, integrating factors of the form mu(x,y)/(y)^n, only the singular cases trying symmetries linear in x and y(x) `, `-> Computing symmetries using: way = 3 `, `-> Computing symmetries using: way = exp_sym Try integration with the canonical coordinates of the symmetry [0, y] -> Calling odsolve with the ODE`, diff(_b(_a), _a) = -(_b(_a)^3-_a^2*_b(_a)-_a)/_b(_a), _b(_a), explicit` *** Sublevel 2 *** 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 trying Abel Looking for potential symmetries Looking for potential symmetries Looking for potential symmetries trying inverse_Riccati trying an equivalence to an Abel ODE differential order: 1; trying a linearization to 2nd order --- trying a change of variables {x -> y(x), y(x) -> x} differential order: 1; trying a linearization to 2nd order trying 1st order ODE linearizable_by_differentiation trying differential order: 2; exact nonlinear -> Calling odsolve with the ODE`, -_b(_a)^2*_a^2+(diff(_b(_a), _a))^2+c__1 = 0, _b(_a)` *** Sublevel 2 *** Methods for first order ODEs: -> Solving 1st order ODE of high degree, 1st attempt trying 1st order WeierstrassP solution for high degree ODE trying 1st order WeierstrassPPrime solution for high degree ODE trying 1st order JacobiSN solution for high degree ODE trying 1st order ODE linearizable_by_differentiation trying differential order: 1; missing variables trying simple symmetries for implicit equations Successful isolation of d_b/d_a: 2 solutions were found. Trying to solve each resulting ODE. *** Sublevel 3 *** Methods for first order ODEs: --- Trying classification methods --- trying homogeneous types: trying exact Looking for potential symmetries trying an equivalence to an Abel ODE trying 1st order ODE linearizable_by_differentiation -> Solving 1st order ODE of high degree, Lie methods, 1st trial `, `-> Computing symmetries using: way = 3 `, `-> Computing symmetries using: way = 4 `, `-> Computing symmetries using: way = 2 `, `-> Computing symmetries using: way = 2 -> Solving 1st order ODE of high degree, 2nd attempt. Trying parametric methods trying dAlembert -> Calling odsolve with the ODE`, diff(y(x), x) = y(x)*x^2/((x^2+c__1)^(1/2)*y(x)^2+x^3+c__1*x), y(x)` *** Sublevel 3 *** 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 trying exact Looking for potential symmetries trying inverse_Riccati trying an equivalence to an Abel ODE differential order: 1; trying a linearization to 2nd order --- trying a change of variables {x -> y(x), y(x) -> x} differential order: 1; trying a linearization to 2nd order trying 1st order ODE linearizable_by_differentiation -> Calling odsolve with the ODE`, diff(y(x), x) = x/((x^2+c__1)^(1/2)*y(x)*(x+(x^2+c__1)^(1/2)/y(x)^2)), y(x)` *** Sublevel 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 trying exact Looking for potential symmetries trying inverse_Riccati trying an equivalence to an Abel ODE differential order: 1; trying a linearization to 2nd order --- trying a change of variables {x -> y(x), y(x) -> x} differential order: 1; trying a linearization to 2nd order trying 1st order ODE linearizable_by_differentiation -> Solving 1st order ODE of high degree, Lie methods, 2nd trial `, `-> Computing symmetries using: way = 4 `, `-> Computing symmetries using: way = 5 `, `-> Computing symmetries using: way = 5 trying 2nd order, integrating factor of the form mu(x,y) -> trying 2nd order, the S-function method *** Sublevel 2 *** Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear trying Bernoulli <- Bernoulli successful *** Sublevel 2 *** Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear <- 1st order linear successful -> Calling odsolve with the ODE`, diff(y(x), x) = (y(x)^2*x^2+c__1)^(1/2), y(x)` *** Sublevel 2 *** Methods for first order ODEs: --- Trying classification methods --- trying homogeneous types: differential order: 1; looking for linear symmetries trying exact Looking for potential symmetries trying an equivalence to an Abel ODE trying 1st order ODE linearizable_by_differentiation --- Trying Lie symmetry methods, 1st order --- `, `-> Computing symmetries using: way = 3 `, `-> Computing symmetries using: way = 5 trying symmetry patterns for 1st order ODEs -> 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 symmetry patterns of the forms [F(x),G(y)] and [G(y),F(x)] -> trying a symmetry pattern of the form [F(x),G(x)] -> trying a symmetry pattern of the form [F(y),G(y)] -> 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 a symmetry pattern of conformal type -> trying 2nd order, No Point Symmetries Class V -> trying 2nd order, No Point Symmetries Class V -> trying 2nd order, No Point Symmetries Class V trying 2nd order, integrating factor of the form mu(x,y)/(y)^n, only the general case -> trying 2nd order, dynamical_symmetries, only a reduction of order through one integrating factor of the form G(x,y)/(1+H(x,y)*y)^ --- Trying Lie symmetry methods, 2nd order --- `, `-> Computing symmetries using: way = 3`[0, y]
✗ Solution by Maple
dsolve(diff(y(x),x)*diff(diff(y(x),x),x)-x^2*y(x)*diff(y(x),x)-x*y(x)^2=0,y(x), singsol=all)
\[ \text {No solution found} \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[-(x*y[x]^2) - x^2*y[x]*y'[x] + y'[x]*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
Not solved