Question

dy/dt=x dx/dt=9y,then y=

Answer 1

SOLUTION

GIVEN

$\displaystyle \sf{ \frac{dy}{dt} = x \: \: , \: \frac{dx}{dt} = 9y}$

TO DETERMINE

The value of y

EVALUATION

Here the given system of Differential equations are

$\displaystyle \sf{ \frac{dy}{dt} = x \: \: \: \: \: \: - - - (1)}$

$\displaystyle \sf{ \frac{dx}{dt} = 9y \: \: \: - - - - (2)}$

Differentiating both sides of Equation 1 we get

$\displaystyle \sf{ \frac{ {d}^{2} y}{d {t}^{2} } = \frac{dx}{dt} }$

$\displaystyle \sf{ \implies \: \frac{ {d}^{2} y}{d {t}^{2} } = 9y} \: \: \: - - - - (3)$

$\sf{Let \: \: y = {e}^{mt} \: \: be \: the \: trial \: solution }$

Then the auxiliary equation is

$\sf{ {m}^{2} = 9}$

$\sf{ \implies \: {m}^{2} - 9 = 0}$

$\sf{ \implies \: m = \pm \: 3}$

Hence the required solution is

$\sf{y = a {e}^{3t} + b {e}^{ - 3t} }$

Where a and b are arbitrary constants

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