# You are given general sol'n - find differential equation

• accountkiller
In summary, you found the derivatives of the constants, and then used that information to solve for y'' + 10y' = 0. You then found the a, b, and c coefficients of the differential equation.
accountkiller

## Homework Statement

You are given a solution - find the homogeneous, 2nd order differential equation ay'' + by' + cy = 0

## Homework Equations

y(x) = c1 + c2e-10x

## The Attempt at a Solution

I found the derivatives first.
y'(x) = c1 - 10c2e-10x
y''(x) = c1 + 100c2e-10x

But I have no idea where to go from there. The answer is y'' + 10y' = 0. Looking at that answer, yes I see why that is the answer, but what work did I have to show to get to that answer?

I'd appreciate any guidance, thanks!

Last edited:

## Homework Statement

You are given a solution - find the homogeneous, 2nd order differential equation ay'' + by' + cy = 0

## Homework Equations

y(x) = c1 + c2e-10x

## The Attempt at a Solution

I found the derivatives first.
y'(x) = c1 - 10c2e-10x
y''(x) = c1 + 100c2e-10x
For starters, the derivative of any constant (e.g., c1) is 0.
But I have no idea where to go from there. The answer is y'' + 10y' = 0. Looking at that answer, yes I see why that is the answer, but what work did I have to show to get to that answer?

I'd appreciate any guidance, thanks!

From the solution you are given, e0x (= 1) and e-10x are basic solutions. These have something to do with the characteristic equation.

Ah, duh! Right, constants ' = 0, so y'(x) = -10c2e-10x and y''(x) = 100c2e-10x.

So I understand that e0x (=1) and e-10x are solutions. But how do I involve the characteristic equation?
ar2 + br + c = 0

How do I find out the a, b, c? Are they those particular solutions? So a = e0x (=1) and b = e-10x and c = ... ? 0?

Since the book says we are supposed to guess y1 = er1x and y2 = er2x, that means my r1 = 0 and my r2 = -10.

Now.. where to go from there?

OK, good. These are the roots of the characteristic equation ar2 + br + c = 0. And a, b, and c are the coefficients of the diff. equation ay'' + by' + cy = 0. Can you continue from here?

If $r_1$ and $r_2$ are roots of a quadratic equation, then the equation is of the form $a(x- r_1)(x- r_2)= 0$ for some number a.

Ah, so if r1 = and r2 = -1 are the roots of ar2 + br + c = 0, then I have (r + 0)(r + 10) = 0 so my characteristic equation is r2 + 10r = 0, so (1) and (10) are my a and b constants, so putting it in ay'' + by' + cy = 0 gives me the differential equation (1)y'' + 10(y') = 0.

I'm usually good at math but this chapter has, for some reason, been just difficult for me to 'click' with whatever I'm supposed to be understanding. Thanks for all of your help Mark, I really appreciate it!

## 1. What is a general solution in differential equations?

A general solution in differential equations is a formula or equation that represents a family of solutions that satisfy the given differential equation. It contains one or more arbitrary constants that can take on any value, allowing for an infinite number of solutions.

## 2. How do you find the differential equation given a general solution?

To find the differential equation given a general solution, you need to differentiate the general solution with respect to the independent variable. This will result in an equation that contains the dependent variable and its derivatives. The resulting equation is the differential equation that represents the given general solution.

## 3. Can a general solution be unique?

Yes, a general solution can be unique if the differential equation only has one solution. This means that there is no need for arbitrary constants in the general solution, making it a specific solution instead.

## 4. What is the importance of finding the differential equation given a general solution?

Finding the differential equation given a general solution allows for a deeper understanding of the relationship between the independent and dependent variables. It also helps in predicting future behavior and making more accurate mathematical models.

## 5. Are there any techniques for finding the differential equation given a general solution?

Yes, there are various techniques for finding the differential equation given a general solution. One method is to use the method of undetermined coefficients, where you assume a specific form for the solution and solve for the coefficients. Another approach is using the method of variation of parameters, where you solve for a particular solution using a linear combination of known solutions and a new function.

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