Internal problem ID [10143]
Internal file name [OUTPUT/9090_Monday_June_06_2022_06_29_00_AM_50071104/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1822 (book 6.231).
ODE order: 2.
ODE degree: 1.
The type(s) of ODE detected by this program : "second_order_integrable_as_is"
Maple gives the following as the ode type
[[_2nd_order, _exact, _nonlinear], [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_poly_yn]]
Unable to solve or complete the solution.
\[ \boxed {\left (2 y^{2} y^{\prime }+x^{2}\right ) y^{\prime \prime }+2 y {y^{\prime }}^{3}+3 x y^{\prime }+y=0} \]
Integrating both sides of the ODE w.r.t \(x\) gives \begin {align*} \int \left (\left (2 y^{2} y^{\prime }+x^{2}\right ) y^{\prime \prime }+\left (2 y {y^{\prime }}^{2}+3 x \right ) y^{\prime }+y\right )d x &= 0 \\ {y^{\prime }}^{2} y^{2}+y^{\prime } x^{2}+x y = 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 }&=\frac {-x^{2}+\sqrt {-4 y^{3} x +x^{4}+4 y^{2} c_{1}}}{2 y^{2}} \tag {1} \\ y^{\prime }&=-\frac {x^{2}+\sqrt {-4 y^{3} x +x^{4}+4 y^{2} c_{1}}}{2 y^{2}} \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 \[ \left (2 y^{2} y^{\prime }+x^{2}\right ) y^{\prime \prime }+\left (2 y {y^{\prime }}^{2}+3 x \right ) y^{\prime }+y = 0 \] Integrating both sides of the ODE w.r.t \(x\) gives \begin {align*} \int \left (\left (2 y^{2} y^{\prime }+x^{2}\right ) y^{\prime \prime }+\left (2 y {y^{\prime }}^{2}+3 x \right ) y^{\prime }+y\right )d x &= 0 \\ {y^{\prime }}^{2} y^{2}+y^{\prime } x^{2}+x y = 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 }&=\frac {-x^{2}+\sqrt {-4 y^{3} x +x^{4}+4 y^{2} c_{1}}}{2 y^{2}} \tag {1} \\ y^{\prime }&=-\frac {x^{2}+\sqrt {-4 y^{3} x +x^{4}+4 y^{2} c_{1}}}{2 y^{2}} \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.
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) trying differential order: 2; exact nonlinear -> Calling odsolve with the ODE`, (diff(_b(_a), _a))^2*_b(_a)^2+_a^2*(diff(_b(_a), _a))+_b(_a)*_a+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) = (-2*y(x)^2*x^3+(-4*y(x)^2*x^3-4*c__1*x+y(x)^2)^(1/2)*y(x)+2*c__1*x-y(x)^2)/(x*( 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 -> Calling odsolve with the ODE`, diff(y(x), x) = 2*(y(x)^2*x^3+2*c__1*x^2-y(x)^2+(-4*y(x)^2*x^3-4*c__1*x^2+y(x)^2)^(1/2)*y(x))/( 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, 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) differential order: 2; looking for linear symmetries differential order: 2; found: 1 linear symmetries. Trying reduction of order `, `2nd order, trying reduction of order with given symmetries:`[x, y]
✗ Solution by Maple
dsolve((2*y(x)^2*diff(y(x),x)+x^2)*diff(diff(y(x),x),x)+2*y(x)*diff(y(x),x)^3+3*x*diff(y(x),x)+y(x)=0,y(x), singsol=all)
\[ \text {No solution found} \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y[x] + 3*x*y'[x] + 2*y[x]*y'[x]^3 + (x^2 + 2*y[x]^2*y'[x])*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
Not solved