Question

Solve simultaneous equation dx/dt=3y dy/dt=3x

Answer 1

SOLUTION

TO DETERMINE

Solve the simultaneous equations

$\displaystyle \sf{ \frac{dx}{dt} = 3y \: \: \: \: \: and \: \: \: \: \frac{dy}{dt} = 3x}$

EVALUATION

Here the given simultaneous equations are

$\displaystyle \sf{ \frac{dx}{dt} = 3y \: \: \: \: \: \: - - - - - (1)}$

$\displaystyle \sf{\frac{dy}{dt} = 3x \: \: \: \: \: \: \: - - - - (2)}$

Differentiating equation 1 with respect to t we get

$\displaystyle \sf{ \frac{ {d}^{2} x}{d {t}^{2} } = 3\frac{dy}{dt} }$

$\displaystyle \sf{ \implies \: \frac{ {d}^{2} x}{d {t}^{2} } = 3 \times 3x}$

$\displaystyle \sf{ \implies \: \frac{ {d}^{2} x}{d {t}^{2} } = 9x}$

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

So the auxiliary equation is

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

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

$\therefore \: \sf{x = a {e}^{3t} + b {e}^{ - 3t} }$

Equation 1 gives

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

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

Hence the required solution is

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

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

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Answer 2

Answer:

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