Fourier transform of differentials equation

[naspek]

should i test for e^-ax or x^5 or e^(x^2) or
<br /> y(x)=\int_{-\infty}^{\infty} e^{-a|x|}f(x-t)dt<br />
i'm sorry coz keep on asking.. but, i really don't know which one should i use to run the test.

 

[gabbagabbahey]

You want to test to see if y(x) converges for each of your two f(x)'s. If the integral doesn't converge, then no solution exists.

 

[naspek]

You want to test to see if y(x) converges for each of your two f(x)'s. If the integral doesn't converge, then no solution exists.

so.. i need to test the equations below?
<br /> y(x)=\int_{-\infty}^{\infty} e^{-a|x|}f(x-t)^{5}dt<br />

and

<br /> y(x)=\int_{-\infty}^{\infty} e^{-a|x|}f(e^{(x-t)^{2}})dt<br />

correct?

 

[naspek]

You want to test to see if y(x) converges for each of your two f(x)'s. If the integral doesn't converge, then no solution exists.

by performing the improper integral test, when f(x) = x^5, the integral will converges
when f(x) = e^(x^2), the integral will diverges..

hence.. when f(x) = x^5, the is a solution, when f(x) = e^(x^2), there is no solution..
correct?

 

[gabbagabbahey]

I just noticed that you have an error in your expression. Convolution tells you that y(x)=\frac{1}{2a}\int_{-\infty}^{\infty}f(x-t)e^{-a|t|}dt, not \frac{1}{2a}\int_{-\infty}^{\infty}f(x-t)e^{-a|x|}dt.

 

[naspek]

I just noticed that you have an error in your expression. Convolution tells you that y(x)=\frac{1}{2a}\int_{-\infty}^{\infty}f(x-t)e^{-a|t|}dt, not \frac{1}{2a}\int_{-\infty}^{\infty}f(x-t)e^{-a|x|}dt.

change it already.. thanks

 

[naspek]

the first part of the general solution should be c1(cos XXXX) + c1(sin XXXX) right?
just ignore the XXXX.. just want to know the pattern..