What is a solution to the differential equation $(\frac{d y}{d x}) - y - e^{3 x} = 0$ $(dy/dx) - y - e^(3x) = 0$ ?

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Answer 1 · 2016-07-11T20:50:52

$y = \frac{1}{2} e^{3 x} + C e^{x}$ $y = 1/2 e^(3x) + Ce^x$

$\frac{d y}{d x} - y - e^{3 x} = 0$ $dy/dx - y - e^(3x) = 0$

this is not separable so we use an Integrating Factor (IF)

$\frac{d y}{d x} - y = e^{3 x}$ $dy/dx - y = e^(3x)$

$I F = e^{\int (- 1) d x} = e^{- x}$ $IF = e^{int (-1) dx} = e^-x$

$\Rightarrow e^{- x} \frac{d y}{d x} - e^{- x} y = e^{2 x}$ $implies e^-x dy/dx - e^-x y = e^(2x)$

Or $\frac{d}{d x} (e^{- x} y) = e^{2 x}$ $d/dx ( e^-x y )= e^(2x)$

$e^-x y = int \ e^(2x) \ dx$

$e^-x y = 1/2 e^(2x) + C$

$y = 1/2 e^(3x) + Ce^x$

What is a solution to the differential equation (dy/dx) - y - e^(3x) = 0(dydx)−y−e3x=0(dy/dx) - y - e^(3x) = 0?