Question

find the general solution of the system dx/dt=x and dy/dt=y​

Answer 1

Given:

dx/dt=x and dy/dt=y

To Find:

find the general solution of the system dx/dt=x and dy/dt=y​?

Solution:

firstly, consider dx/dt=x

$\dfrac{dx}{dt}=x\\\\\dfrac{1}{x}dx=dt\\\\\int\dfrac{1} {x} \, dx=\int {1} \, dt \\\\ln(x)=t+c1$  

$x=e^{t+c1}=e^t.e^{c1}\\x=Ae^t$   ....(1)   (where, $e^{c1}=A=constant$)

Now, consider dy/dt=y

similarly,

$ln(y)=t+c2$

$\Rightarrow y = B e^t$  ---(2)   (where, $e^{c2}=B= constant$)

therefore, general solution of dx/dt=x and dy/dt=y are $x=Ae^t$ and $y = Be^t$ respectively.

Hence, from equation (1) and (2) , we get

$x = \dfrac{A}{B} \ y\\\\x=Cy$ where, C is the constant A/B .

Hence, x=Cy is the general solution of the system dx/dt=x and dy/dt=y.