- #1

Safder Aree

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- Homework Statement
- $$ u_{tt}(x,t) + 2u_t(x,t) = -u(x,t), -\infty < x < \infty, t> 0$$

$$u(x,t), as |x| \rightarrow \infty, t>0$$

$$u(x,0) = f(x), u_t(x,0)= g(x) , -\infty < x < \infty $$

- Relevant Equations
- Fourier Transform equation:

$$ \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} f(x) e^{iwx} dx $$

I am struggling to figure out how to approach this problem. I've only solved a homogenous heat equation $$u_t = u_{xx}$$ using a Fourier transform, where I can take the Fourier transform of both sides then solve the general solution in Fourier terms then inverse transform. However, since this question has extra terms I'm a little confused.$$ u_{tt}(x,t) + 2u_t(x,t) = -u(x,t), -\infty < x < \infty, t> 0$$

$$u(x,t), as |x| \rightarrow \infty, t>0$$

$$u(x,0) = f(x), u_t(x,0)= g(x) , -\infty < x < \infty $$

Any advice and or guidance would be greatly appreciated. All I would do is take the Fourier transform of all the terms but from there I don't think I know what to do.`

$$u(x,t), as |x| \rightarrow \infty, t>0$$

$$u(x,0) = f(x), u_t(x,0)= g(x) , -\infty < x < \infty $$

Any advice and or guidance would be greatly appreciated. All I would do is take the Fourier transform of all the terms but from there I don't think I know what to do.`