Question

Solve the differential equation d^2y/dx^2-3dy/dx-4y=0​

Answer 1

SOLUTION

TO SOLVE

The differential equation

$\displaystyle\sf{ \frac{ {d}^{2}y }{d {x}^{2} } - 3 \frac{dy}{dx} - 4y = 0}$

EVALUATION

Here the given differential equation is

$\displaystyle\sf{ \frac{ {d}^{2}y }{d {x}^{2} } - 3 \frac{dy}{dx} - 4y = 0}$

Let

$\displaystyle\sf{ y = {e}^{mx} }$

be the trial solution

Then the auxiliary equation is

$\displaystyle\sf{ {m}^{2} - 3m - 4 = 0}$

$\displaystyle\sf{ \implies {m}^{2} - 4m + m - 4 = 0}$

$\displaystyle\sf{ \implies (m + 1)(m - 4)= 0}$

$\displaystyle\sf{ \implies m = - 1 \: ,\: 4}$

Hence the required solution is

$\sf{y = a \: {e}^{ - x} + b \: {e}^{4x} }$

Where a and b are constants

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