(x^2) y'' + p x y' + q y = 0 Euler's equidimensional equation where p and q are constants. Show...

Question:

The equation {eq}(x^2)y''+pxy'+qy=0 {/eq} where {eq}p {/eq} and {eq}q {/eq} are constants, is called Euler's equidimensional equation. Show that the change of {eq}x=e^t {/eq} transforms Eulers equation into a new one with constant coefficients. Apply this to find the general solution of {eq}(x^2)y''+2xy'-12y=0 {/eq}

Euler's Equidimensional Equation:

The equation {eq}(x^2)y''+pxy'+qy=0 {/eq} where {eq}p {/eq} and {eq}q {/eq} are constants, is called Euler's equidimensional equation. There are two different approaches to solving this equation. First, we can define {eq}x = e^t, {/eq} and apply the Chain Rule to transform the equation into a differential equation with {eq}y {/eq} as a function of {eq}t. {/eq} Second, we can make the assumption that {eq}y = x^r {/eq} for some undetermined value(s) of {eq}r, {/eq} then differentiate twice and substitute into the equation.

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Since {eq}x = e^t {/eq} and {eq}y = y(x), {/eq} we have {eq}y = y(e^t). {/eq} Applying the Chain Rule gives

{eq}\displaystyle\frac{dy}{dt} = e^t...

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First-Order Linear Differential Equations

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Chapter 16 / Lesson 3
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In this lesson you'll learn how to solve a first-order linear differential equation. We first define what such an equation is, and then we give the algorithm for solving one of that form. Specific examples follow the more general description of the method.


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