Solving this Differential Equation using Convolution

[Jack1235]

Homework Statement

The differential equation $5s(t)-4s''(t)=r(t)$ can be solved by the convolution $s=q*r$ where $q(t)=c_1*\exp(-c_2*|(t)|)$. Find $c_1+c_2$.

Relevant Equations

$$\int_{-\infty}^{\infty} r(t-u)s(u) \,du$$

$s=c_1*\exp(-c_2*|(t)|)*r(t)$ But how can I solve $c_1+c_2$ ?

 

[BvU]

Hello Jack,

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I am very much afraid your post will be deleted by one of the mentors soon: it doesn't comply with the PF rules and guidelines (there is no posted apparent effort from your side).
Before that: $\LaTeX$ has to be enclosed by double $$ (for displayed math) and double $for in-line math: The differential equation 5s(t)-4s''(t)=r(t) can be solved by the convolution s=q*r where q(t)=c_1*\exp(-c_2*|(t)|). Find c_1+c_2 Relevant Equations: \left (\int_{-\infty}^{\infty} r(t-u)s(u) \,du\right ) s=c_1*\exp(-c_2*|(t)|)*r(t) But how can I solve c_1+c_2 ? With$ it looks better:

Homework Statement: The differential equation $5s(t)-4s''(t)=r(t)$ can be solved by the convolution $s=q*r$ where $q(t)=c_1*\exp(-c_2*|(t)|)$. Find $c_1+c_2$.​

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Relevant Equations: $\int_{-\infty}^{\infty} r(t-u)s(u) \,du$​

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$s=c_1*\exp(-c_2*|(t)|)*r(t)$ But how can I solve $c_1+c_2$ ?​

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$\$​

 

[Jack1235]

So this is my attempt: $5(q*r)-4(q*r)''=r(t)$ and $5(C_1\exp(-C_2|t|)*r)-4*(C_1*\exp(-C_2*|t|)*r)''=r(t)$
In addition to that: $(q*r)''=(-i*2*\pi*\widehat{ts}(\nu))'$

 

[Mark44]

I am very much afraid your post will be deleted by one of the mentors soon: it doesn't comply with the PF rules and guidelines (there is no posted apparent effort from your side).

Since the OP has (belatedly) shown some effort, no moderator action has been taken here.

 

[BvU]

Good. Now, do I have to sort out when a * stands for multiplication and when it means convolution ?

I also find it strange the problem statement asks for $c_1 + c_2$. It seems to me they have different dimensions and can not be added. Could it be the problem composer means to ask for $c_1$ and $c_2$ ?

Furthermore: the relevant equation does not really look like an equation at all.
I suppose you meant $$s = q*r \ \ \Leftrightarrow \ \ s(t) =\int_0^t q(\tau) r(t-\tau) \, d\tau \ \ ?$$

Could you please also explain what you do in post #3 ?

$\$

 

[vela]

Note: I tweaked the first post to get most of the LaTeX to render.

 

[BvU]

Then post #2 now looks pretty dumb ...