(sin x)y''' − 3xy'' + 2y = tan x

February 4, 2016 - 4:01pm

#1
find the interval: third order differential equation

how to find the lateral area of a regular pyramid section modulus of rectangular section what is the equation for pythagorean theorem altitudes in geometry brahmagupta formula proof diagonals in a decagon integral calculus problems and solutions define friction physics herrons formula definition of pentagonal prism tsa of a cone statics supports depreciation formula math transformation formula trigonometry method of integrating factor geometric progression nth term casio engineering calculator basic trig integrals foe beams bending moment of cantilever beam formula resultant force questions answers deflection of a simply supported beam definition of shearing stress calculation of moment of inertia surface area of a cone calculator with slant height radii of gyration the center of gravity of a triangle integral calculus pdf trigonometric functions problems and solutions curvilinear equation conjugate calculus standard parabola equation diagonal of a rectangular prism formula centroid of a right trapezoid cyclic quad right circular cylinder volume all formulae of integration shear stress equation torque seperable differential equations determining moment of inertia finding moments of inertia pythagorean trigonometric identities how to integrate sqrt 1 x 2 what is torsional stress formulas in analytic geometry wide flange section algebra word problems calculator hexahedron volume formula solid mensuration problems with answers hydrostatic formula moment of inertia hollow beam engineering mechanics solutions properties of medians of triangles circumscribed triangle formula formula of a hemisphere equation of tangent of a circle bernoulli equation solver volume of an equilateral triangle volume of a hollow cone surface area of cone without base trigonometric formulaes differentiate equation calculator geometric definition of a parabola prc board exams what is cone in maths prove derivative of cosx altitudes of a triangle derivatives of the inverse trigonometric functions truss cantilever what is altitude geometry quadrilateral formulas hollow sphere formula statics and rigid bodies find quadratic equation from roots formulas of triangle formula for circumference of a sphere newton's law of cooling isosceles trapezoid volume formula

You are interested in listing the interval of validities of the third-order differential equation $(\sin x)y'''-3xy''+2y = \tan x$.That explains why you didn't specify initial condition like $y(0) = 4$.

Here's how to get it...

Modifying the differential equation to look like this:

$$y'''+p(x)y''+q(x)y'+r(x)y = g(x)$$

It becomes...

$$(\sin x)y'''-3xy''+2y = \tan x$$ $$\frac{(\sin x)y'''-3xy''+2y = \tan x}{\sin x}$$ $$y'''-\frac{3x}{\sin x} y'' + \frac{2}{\sin x}y = \frac{\tan x}{\sin x}$$ $$y'''-\frac{3x}{\sin x} y'' + \frac{2}{\sin x}y = \frac{1}{\cos x}$$

To determine the intervals, we need to see where the $p(x)$, $r(x)$, and $g(x)$ to be continuous. We must avoid those points that render $p(x)$, $r(x)$, and $g(x)$ to be discontinuous. Those points we must avoid are:

Setting $\sin x$ to be zero makes $p(x)$ and $r(x)$ to be discontinuous. Sine function goes to zero every $n\pi $ interval.

$$\sin x = 0 \implies x = \color{red}{n}\pi\,, n \in \mathbb Z$$

Setting $\cos x$ to be zero makes $g(x)$ to be discontinuous. Cosine function goes to zero every $(2n+1)\frac{\pi}{2}$ intervals.

$$\cos x =0 \implies x =(2n+1)\color{blue}{\frac{\pi}{2}}\,, n\in \mathbb Z$$

Therefore, the intervals of the differential equation $(\sin x)y'''-3xy''+2y = \tan x$ would be...

$$\mathbb R \setminus \{n\pi \mid \, n \in \mathbb Z\} \setminus \{(2n+1)\frac{\pi}{2} \mid \, n \in \mathbb Z\}$$

It refers to the fact that the solution set is $\mathbb R$ minus than the values $2nπ$ and $(2n+1)\frac{π}{2}$. It is the difference of sets. It an has interval ranging from $-\infty$ to $\infty$, excluding $2nπ$ and $(2n+1)\frac{π}{2}.$