Bt Bt

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Find the general solution of the ordinary differential equation?

y'' - (b^2)y=e^(bt)

Where b is a non negative constant. Consider the cases; b=0 and b>0 separately.

Case 1: b = 0.
The DE reduces to y'' = 1.

Integrate both sides:
y = x + C₁

Integrate both sides again:
y = (1/2)x^2 + C₁ x + C₂.
-------------------------------
Case 2: b > 0.

Homogeneous solution:
The characteristic equation r^2 - b^2 = 0 has roots r = -b, b.
So, y_h = C₁ e^(bt) + C₂ e^(-bt).
----
Particular integral:
Due to the homogeneous solution, assume that y_p = Ate^(bt).

Differentiating yields
y' = Ae^(bt) + bAte^(bt) = Ae^(bt) (bt + 1).
y'' = bAe^(bt) (bt + 1) + Ae^(bt) * b = Ae^(bt) (b^2 t + 2b).

Substitute this into the DE:
Ae^(bt) (b^2 t + 2b) - b^2 * Ate^(bt) = e^(bt)
==> (2b) A = 1
==> A = 1/(2b).

So, the general solution is
y = C₁ e^(bt) + C₂ e^(-bt) + (1/(2b)) te^(bt).

I hope this helps!

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