# Differential Equation: x^2y''-xy'-3y=2x^-(3/2)

• Ric-Veda
In summary, you can use both the variation of parameters and the method of undetermined coefficients to solve a Cauchy-Euler equation.
Ric-Veda

## The Attempt at a Solution

I am not asking to find the answer, just wanted to know whether to use the variation of parameters or undetermined coefficients. Because this was on a test problem and I used variation of parameters instead. I know it is a Cauchy-Euler equation, but do you use the method of undetermined coefficients or variation of parameters?

First, you want to put your equation in standard form as such:

##y''-\frac{1}{x}y'-\frac{3}{x^2}y=2x^{-\frac{7}{2}}##.

Then observe that the terms on the left side decrease in power by an increment of one. This implies that you want solutions ##y## of a form where ##y## is a function of ##x## of some power ##k##, so that you can say that ##y'## is in the ##k-1## power. Then you have ##y''## in the ##k-2## power.

For example, for ##y=ax^{k}##, ##y'=akx^{k-1}##, and ##y''=ak(k-1)x^{k-2}##.

Plug these values in the differential equation, and you have:

##ak(k-1)x^{k-2}-\frac{1}{x}(akx^{k-1})-\frac{3}{x^2}ax^{k}=ax^{k-2}(k(k-1)-k-3)=2x^{-\frac{7}{2}}##.

Now everything is in terms of the same power. I don't know how this would work for non-homogeneous equations, though. I don't exactly know what this method is called, in any case. But this is what I would have done.

Last edited:
Ric-Veda said:

## The Attempt at a Solution

I am not asking to find the answer, just wanted to know whether to use the variation of parameters or undetermined coefficients. Because this was on a test problem and I used variation of parameters instead. I know it is a Cauchy-Euler equation, but do you use the method of undetermined coefficients or variation of parameters?
You can use both. Show your work.

Unless the question specifically tells you to use a particular method, any valid method should be acceptable.

## 1. What is a differential equation?

A differential equation is a mathematical equation that describes the relationship between a function and its derivatives. It involves variables, constants, and derivatives of the function.

## 2. What is the order of a differential equation?

The order of a differential equation is the highest derivative present in the equation. In this case, the order is 2 because the highest derivative is y''.

## 3. How do you solve a differential equation?

There are various methods for solving differential equations, such as separation of variables, substitution, and using integrating factors. In this particular equation, we can use the method of undetermined coefficients to find a particular solution.

## 4. What is the solution to this specific differential equation?

The general solution to this differential equation is y = C1x^(-3/2) + C2x^2, where C1 and C2 are constants. This can be found by substituting the function y = x^m into the equation and solving for m.

## 5. How can differential equations be applied in real life?

Differential equations are used in various fields such as physics, engineering, economics, and biology to model and understand the behavior of systems. For example, they can be used to predict the growth of a population, the movement of a pendulum, or the flow of electricity in a circuit.

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