Fourier transform of differentials equation

[naspek]

should i test for e^-ax or x^5 or e^(x^2) or
$y(x)=\int_{-\infty}^{\infty} e^{-a|x|}f(x-t)dt$
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?
$y(x)=\int_{-\infty}^{\infty} e^{-a|x|}f(x-t)^{5}dt$

and

$y(x)=\int_{-\infty}^{\infty} e^{-a|x|}f(e^{(x-t)^{2}})dt$

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..