Internal problem ID [4165]
Internal file name [OUTPUT/3658_Sunday_June_05_2022_10_05_26_AM_85528302/index.tex]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 31
Problem number: 931.
ODE order: 1.
ODE degree: 2.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[_rational, [_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]
Unable to solve or complete the solution.
\[ \boxed {x \left (-x^{2}+1\right ) {y^{\prime }}^{2}-2 \left (-x^{2}+1\right ) y y^{\prime }+x \left (1-y^{2}\right )=0} \] 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 {y x^{2}+\sqrt {-x^{2} y^{2}+x^{4}+y^{2}-x^{2}}-y}{x \left (x^{2}-1\right )} \tag {1} \\ y^{\prime }&=-\frac {-y x^{2}+\sqrt {-x^{2} y^{2}+x^{4}+y^{2}-x^{2}}+y}{x \left (x^{2}-1\right )} \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.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x \left (-x^{2}+1\right ) {y^{\prime }}^{2}-2 \left (-x^{2}+1\right ) y y^{\prime }+x \left (1-y^{2}\right )=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \left [y^{\prime }=\frac {y x^{2}+\sqrt {-x^{2} y^{2}+x^{4}+y^{2}-x^{2}}-y}{x \left (x^{2}-1\right )}, y^{\prime }=-\frac {-y x^{2}+\sqrt {-x^{2} y^{2}+x^{4}+y^{2}-x^{2}}+y}{x \left (x^{2}-1\right )}\right ] \\ \bullet & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=\frac {y x^{2}+\sqrt {-x^{2} y^{2}+x^{4}+y^{2}-x^{2}}-y}{x \left (x^{2}-1\right )} \\ \bullet & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=-\frac {-y x^{2}+\sqrt {-x^{2} y^{2}+x^{4}+y^{2}-x^{2}}+y}{x \left (x^{2}-1\right )} \\ \bullet & {} & \textrm {Set of solutions}\hspace {3pt} \\ {} & {} & \left \{\mathit {workingODE} , \mathit {workingODE}\right \} \end {array} \]
Maple trace
`Methods for first order ODEs: *** 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 dy/dx: 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 -> Calling odsolve with the ODE`, diff(diff(y(x), x), x)+(diff(y(x), x))/(x*(x^2-1)), y(x)` *** Sublevel 4 *** Methods for second order ODEs: --- Trying classification methods --- trying a quadrature checking if the LODE has constant coefficients checking if the LODE is of Euler type trying a symmetry of the form [xi=0, eta=F(x)] checking if the LODE is missing y <- LODE missing y successful <- 1st order ODE linearizable_by_differentiation successful ------------------- * Tackling next 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 <- 1st order ODE linearizable_by_differentiation successful`
✓ Solution by Maple
Time used: 0.407 (sec). Leaf size: 33
dsolve(x*(-x^2+1)*diff(y(x),x)^2-2*(-x^2+1)*y(x)*diff(y(x),x)+x*(1-y(x)^2) = 0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= -x \\ y \left (x \right ) &= x \\ y \left (x \right ) &= \sqrt {-c_{1}^{2}+1}+\sqrt {x^{2}-1}\, c_{1} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.752 (sec). Leaf size: 75
DSolve[x*(1-x^2)*(y'[x])^2-2*(1-x^2)*y[x]*y'[x]+x*(1-y[x]^2)==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -x \cos \left (2 \tan ^{-1}\left (\sqrt {\frac {x-1}{x+1}}\right )+i c_1\right ) \\ y(x)\to -x \cos \left (2 \tan ^{-1}\left (\sqrt {\frac {x-1}{x+1}}\right )-i c_1\right ) \\ y(x)\to -x \\ y(x)\to x \\ \end{align*}