Eliminating First Derivative in Tricky Differential Equation | Homework Help

In summary, the problem involves a second-order differential equation with a first derivative term. The hint suggests trying to eliminate the first derivative term by using substitutions such as y'=ux or y=ux. Other suggestions include playing around with u = xy' or x(y'2) and u = xy'exp(mx).
  • #1
outhsakotad
32
0

Homework Statement


xy''+2y'+(n^2)*x*y=sin(omega*x)

Hint: Eliminate the first derivative term


Homework Equations





The Attempt at a Solution



I have tried lots of substitutions, but none of them seems to work out. I don't really understand what the hint is getting at. For example, I've tried y'=ux, but then I don't know what to do with the y. I've tried y=ux, but that doesn't seem to help matters at all. And I've tried numerous other silly things that haven't worked out. I'm really out of ideas and stuck. Any little hints would be greatly appreciated.
 
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  • #2
You notice that xy'' + y' looks very much like the derivative of a product.

So I suggest you try playing around with u = xy' or x(y'2); and if that doesn't work maybe u = xy
 

FAQ: Eliminating First Derivative in Tricky Differential Equation | Homework Help

What is a first derivative in a differential equation?

A first derivative in a differential equation is the rate of change of a variable with respect to another variable. It represents the slope of a curve at a specific point and is denoted by dy/dx.

Why would one want to eliminate the first derivative in a tricky differential equation?

In some cases, eliminating the first derivative can make the equation easier to solve or can reveal a simpler solution that may not be apparent with the first derivative present.

How is the first derivative eliminated in a differential equation?

The first derivative can be eliminated by using substitution, integration, or other algebraic manipulations. The specific method used will depend on the structure of the equation and the goal of the elimination.

What are some strategies for tackling tricky differential equations with first derivative elimination?

Some strategies for solving tricky differential equations with first derivative elimination include identifying patterns or relationships within the equation, breaking the equation into smaller parts, and using known formulas and techniques for solving differential equations.

Are there any common mistakes to watch out for when eliminating the first derivative in a differential equation?

Yes, some common mistakes to watch out for when eliminating the first derivative in a differential equation include forgetting to apply the chain rule when substituting for the derivative, making errors in integration, and losing solutions by dividing by zero or neglecting to consider all possible solutions.

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