Internal problem ID [1041]
Internal file name [OUTPUT/1042_Sunday_June_05_2022_01_57_41_AM_93331707/index.tex]
Book: Elementary differential equations with boundary value problems. William F. Trench.
Brooks/Cole 2001
Section: Chapter 2, First order equations. Exact equations. Section 2.5 Page 79
Problem number: 12.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program :
Maple gives the following as the ode type
[NONE]
Unable to solve or complete the solution.
\[ \boxed {y^{2} \sin \left (x \right )+x y^{3} \cos \left (x \right )+\left (\sin \left (x \right ) y x +x y^{3} \cos \left (x \right )\right ) y^{\prime }=0} \] The ode \begin {align*} y^{2} \sin \left (x \right )+x y^{3} \cos \left (x \right )+\left (\sin \left (x \right ) y x +x y^{3} \cos \left (x \right )\right ) y^{\prime } = 0 \end {align*}
is factored to \begin {align*} y \left (y^{2} \cos \left (x \right ) y^{\prime } x +y^{2} \cos \left (x \right ) x +\sin \left (x \right ) y^{\prime } x +\sin \left (x \right ) y\right ) = 0 \end {align*}
Which gives the following equations \begin {align*} y = 0\tag {1} \\ y^{2} \cos \left (x \right ) y^{\prime } x +y^{2} \cos \left (x \right ) x +\sin \left (x \right ) y^{\prime } x +\sin \left (x \right ) y = 0\tag {2} \\ \end {align*}
Each of the above equations is now solved.
Solving ODE (1) Since \(y = 0\), is missing derivative in \(y\) then it is an algebraic equation. Solving for \(y\). \begin {align*} \end {align*}
Solving ODE (2) Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{2} \sin \left (x \right )+x y^{3} \cos \left (x \right )+\left (\sin \left (x \right ) y x +x y^{3} \cos \left (x \right )\right ) y^{\prime }=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-\frac {y^{2} \sin \left (x \right )+x y^{3} \cos \left (x \right )}{\sin \left (x \right ) y x +x y^{3} \cos \left (x \right )} \end {array} \]
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 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 Lie symmetry methods, 1st order --- `, `-> Computing symmetries using: way = 3 `, `-> Computing symmetries using: way = 4 `, `-> 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)] `, `-> Computing symmetries using: way = HINT -> Calling odsolve with the ODE`, diff(y(x), x)+(sin(2*x)-2*y(x))/sin(2*x), y(x)` *** 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)/x, y(x)` *** 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)*sin(2*x)+2*x*sin(2*x)-2*y(x)*x)/(sin(2*x)*x), y(x)` *** 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)-1, y(x)` *** Sublevel 2 *** Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear <- 1st order linear successful `, `-> Computing symmetries using: way = HINT -> Calling odsolve with the ODE`, diff(y(x), x)+y(x)/x, y(x)` *** 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)-(x+1)/x, y(x)` *** Sublevel 2 *** Methods for first order ODEs: --- Trying classification methods --- trying a quadrature <- quadrature successful -> Calling odsolve with the ODE`, diff(y(x), x)-(2*y(x)+x)/x, y(x)` *** Sublevel 2 *** Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear <- 1st order linear successful -> 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`
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 5
dsolve((y(x)^2*sin(x)+x*y(x)^3*cos(x))+(x*sin(x)*y(x)+x*y(x)^3*cos(x))*diff(y(x),x)=0,y(x), singsol=all)
\[ y \left (x \right ) = 0 \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[(y[x]^2*Sin[x]+x*y[x]^3*Cos[x])+(x*Sin[x]*y[x]+x*y[x]^3*Cos[x])*y'[x]==0,y[x],x,IncludeSingularSolutions -> True]
Not solved