- #1

- 508

- 4

Consider the equation,

[tex]\frac{dy}{dx} + ay = Q(x)[/tex]

where a is positive and Q(x) is continuous on [tex][0,\infty ],[/tex]

Show that the general solution to the above equation can be written as..

[tex]y(x) = y(x_{0})e^{-a(x-x_{0})}+e^{-ax}\int^{x}_{x_{0}}e^{at}Q(t)dt[/tex]

Where [tex]x_{0}[/tex] is a non-negative constant

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This part is easy, I just differentiated y(x) and plugged it into the equation. My real struggle is in the next part.

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If [tex]|Q(x)|\leq k for x\geq x_{0}[/tex] where k and [tex]x_{0}[/tex] are non-negative constante, show that..

[tex]|y(x)|\leq |y(x_{0})|e^{-a(x-x_{0})}+\frac{k}{a}[1-e^{-a(x-x_{0})}], for x\geq x_{0}[/tex]

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The only thing I can come up with at the moment is plugging in the expression for y(x) and using the triangle inequality. I don't know where it would lead me though..