# 3rd order differential eqn (wronskian)

• glitchy
In summary, the conversation discusses finding the particular solution y for the equation y''' + 25y' = csc(5x). The person speaking is having minor difficulties and has tried using undetermined coefficients but it did not work for the given function. They then explain the method of variation of parameters and provide the equations needed to solve for the unknown functions. Finally, they provide their solution for y in terms of constants and integrals.
glitchy
y''' + 25y' = csc(5x)

i got the y (complimentary) = C1 + C2cos5x + C3sin5x. I'm just having minor difficulties getting the y (particular).

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Only minor difficulties? I don't see how you can expect anyone to help you if you won't say what you have done. What difficulties have you had? What method did you try?

Since csc(5x) is NOT one of the function that typically satisfy a "homogeneous linear equation with constant coefficients" (exponentials, polynomials, sine and cosine, and combinations of those) you can't "guess" a solution to use "undetermined coefficients". Did you try "variation of parameters"? That is, look for a solution of the form
y(x)= u(x)+ v(x)cos(5x)+ w(x) sin(5x) where u, v, w are unknown functions of x.
(there exist an infinite number of such functions that will satisfy the equation.)

(I presume you meant sin(5x) rather than cos(5x) twice!)

y'= u'+ v'cos(5x)- 5v(x) sin(5x)+ w' sin(5x)+ 5w(x) cos(5x)
Narrow the search to those functions such that
u'+ v'cos(5x)+ w' sin(5x)= 0 so that
y'= -5v(x) sin(5x)+ 5w(x) cos(5x) and
y"= -5v' sin(5x)- 25v(x)cos(5x)+ 5w' cos(5x)- 25 w(x)sin(5x)

Again narrow the search by requiring that
-5v' sin(5x)+ 5w' cos(5x)= 0 so that
y"= -25 v(x) cos(5x)-25w(x) sin(5x) and differentiate once more:

y'''= -25v' cos(5x)+ 125v sin(5x)- 25w' sin(5x)- 125 w cos(5x)

Putting those into the differential equation, because 1, sin(5x), and cos(5x) satisfy the homogenous equation, all terms involving only v and w will cancel leaving a single equation for v' and w'. That, together with the equations
u'+ v'cos(5x)+ w' sin(5x)= 0 and -5v' sin(5x)+ 5w' cos(5x)= 0 give you three equations you can solve for u', v', and w' and then integrate.

so far i got
W = 125
W1 = 5csc5x
W2 = -5cos5xsin5x=-5(cos5x)^2 . tan5x
W3 =-5

therefor

U'1 = csc5x/25
U'2 = -cos5xsin5x/25
U'3 = -1/25

y=C1 + C2cos5x + C3sin5x + (1/5)Intan(5x/2) + (1/250)(cos5x)^3 - (1/25)xsin5x

i got that answer. can anyone verify that for me.

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## 1. What is a 3rd order differential equation?

A 3rd order differential equation is a type of mathematical equation that involves the third derivative of a function. It is commonly used to model physical phenomena in science and engineering.

## 2. What is a Wronskian?

A Wronskian is a mathematical tool used to determine the linear independence of a set of functions. In the context of 3rd order differential equations, the Wronskian is used to determine the general solution of the equation.

## 3. Why is the Wronskian important in 3rd order differential equations?

The Wronskian is important in 3rd order differential equations because it helps to determine whether a set of functions are linearly independent, which is a necessary condition for finding the general solution of the equation.

## 4. How do you solve a 3rd order differential equation using the Wronskian?

To solve a 3rd order differential equation using the Wronskian, you first need to find a set of linearly independent solutions to the equation. Then, you can use the Wronskian to determine the general solution by taking the determinant of the matrix formed by the solutions and their derivatives.

## 5. What are some applications of 3rd order differential equations and the Wronskian?

3rd order differential equations and the Wronskian have various applications in physics, engineering, and other fields of science. They can be used to model the motion of systems with three degrees of freedom, such as oscillating systems. The Wronskian is also useful in solving boundary value problems and eigenvalue problems in quantum mechanics.

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