Given $\dot x(t)=3x(t)$ and $x(0)=\frac32$ what is $x(2)$?

Question

I believe this is a differential delay equation but I'm not sure. I've tried integration but that was confusing. It has also been suggested that I can find the original equation from the first derivative given but that's not right either.

$$\frac{dx}{dt}=3x^2$$ the original would be $x^3$

Answer 1

You have

$$\dot x = \frac{d}{dt}x = 3x$$

multiply $dt$

$$dx = 3xdt$$

divide $x$

$$\frac{1}{x}dx = 3dt$$

integrate

$$\int \frac{1}{x}dx = \int 3dt$$

$$\ln x + C_{apples} = 3t + C_{oranges}$$

make a fruit salad

$$\ln x = 3t + C_{\text{fruit salad}}$$

exponentializify¹

$$x = e^{3t + C_{\text{fruit salad}}}=e^{3t}e^{C_{\text{fruit salad}}}$$

add some chopped chocolate to the salad

$$x = e^{3t}e^{C_{\text{fruit salad}}}=e^{3t}C_{\text{fruit salad with chocolate}}$$

try the salad with $x(0)=\frac32$

$$\frac32 =e^{3\cdot 0}C_{\text{fruit salad with chocolate}}=C_{\text{fruit salad with chocolate}}$$

if you think the salad is appropriately seasoned, serve on 2

$$x(2)=e^{3\cdot 2}C_{\text{fruit salad with chocolate}} = e^6\frac32$$

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¹that is a word

Answer 2

Notice, we have $$x'(t)=3x(t)$$ $$\frac{dx}{dt}=3x$$ $$\frac{dx}{x}=3dt$$ $$\int \frac{dx}{x}=3\int dt$$ $$\ln |x|=3t+C$$$$\iff x=e^{3t+C}$$ Now, setting $x=\frac{3}{2}$ at $t=0$, we get

$$\frac{3}{2}=e^{3(0)+C}\iff e^C=\frac{3}{2}\iff C=\ln\frac{3}{2}$$

Now, setting the value of $C$ we get $$x=e^{3t+\ln\frac{3}{2}}=e^{3t}e^{\ln\frac{3}{2}}=\frac{3}{2}e^{3t}$$

Now, setting $t=2$, we get $$x(2)=\frac{3}{2}e^{3(2)}=\frac{3}{2}e^{6}$$

Hence, we get $$\bbox[5pt, border:2.5pt solid #FF0000]{\color{blue}{x(2)=\frac{3}{2}e^{6}}}$$

Answer 3

$$\frac{dx(t)}{dt} = 3 x(t) \Leftrightarrow x(t) = k\cdot e^{3t}$$

Now the boundary condition:

$$ k\cdot e^0=\frac{3}{2} \Leftrightarrow k=\frac{3}{2}$$

which yields:

$$ x(t) = \frac{3}{2}e^{3t}$$

It's now a matter of calculating this function for $t = 2$:

$$ x(2) = \frac{3}{2} e^6 \approx 605.14$$