Solution to 2nd order diff eq

In summary, the conversation includes a question about finding the general solution to a homogeneous differential equation and the attempt to solve it using the exponential form. However, there is an issue with repeated roots and the suggestion is made to try using general anti-derivatives instead. The person speaking hopes nobody will give away the answer completely.
  • #1
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Never mind, I figured it out.

Here's the question:
Find the general solution to the homogeneous differential equation
https://webwork.math.uga.edu/webwork2_files/tmp/equations/59/540a7a16e5c4e841a098d9d2a72f0a1.png

The solution has the form https://webwork.math.uga.edu/webwork2_files/tmp/equations/06/69c97d88bd2a92e464b45652c75c181.png

enter your answers so that https://webwork.math.uga.edu/webwork2_files/tmp/equations/b6/bf0051fdc1775f5fe9263992f485f41.png

I'm supposed to find f1(t) and f2(t).

I know the form ar^2+br+c=0 but in this case it's only r^2=0 so r=0.
Also y=c1f1(t)+c2f2(t)=c1e^(r1t)+c2e^(r2t)

f1(t)=e^(r1t) and f1(0)=1 so f1(0)=e^(r1*0)=1
I know that's right

I'm stuck on f2(t)
f2(t)=e^(r2t) and f2(2)=0 so f2(2)=e^(r2*2)=2
I tried solving for r2
r2*2=ln(2)
r2=ln(2)/2=.3466
and then plugged it into e^(rt), so it was e^(.3466*t)
This isn't right.

I don't know what else to try. Both r1 and r2 should be equal to 0 to satisfy r^2=0, but then f2(t)=e^(rt)=e^(0*t)=1 and that wouldn't satisfy f(2)=2.

Can anyone tell me what else I can try?

Thanks a lot.
 
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  • #2
There is a problem with your attempt. You have repeated roots. The exponential form doesn't work. If that does not get you anywhere, you can just use general anti-derivatives (integrate both sides) to get the general form of the solution.

This ought to give you the idea of the problem. I hope nobody tries to one-up my answer and gives it all away like you couldn't get it.
 

1. What is a second order differential equation?

A second order differential equation is a mathematical equation that involves a function, its derivatives, and independent variables. It represents a relationship between a function and its rate of change.

2. What is the general form of a second order differential equation?

The general form of a second order differential equation is y'' + p(x)y' + q(x)y = g(x), where y is the dependent variable, x is the independent variable, and p(x), q(x), and g(x) are functions of x.

3. How do you solve a second order differential equation?

To solve a second order differential equation, you can use various methods such as separation of variables, substitution, or the method of undetermined coefficients. The specific method used depends on the type of equation and its initial conditions.

4. Why are second order differential equations important?

Second order differential equations are important because they are used to model many real-world phenomena, including motion, heat transfer, and electrical circuits. They also have various applications in physics, engineering, and other scientific fields.

5. Can second order differential equations have multiple solutions?

Yes, second order differential equations can have multiple solutions. This is because they can have different initial conditions or boundary conditions, which can lead to different solutions. Additionally, some equations may have a general solution that includes a constant, which can take on different values and result in different solutions.

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