I have the DE
$y''-4y'+4y=e^{2x}$
The general solution to the corresponding homogeneous equation is,
$y_h(x)=e^{2x}(A+Bx)$
How do I find a particular solution $y_p(x)$ that fits with this?
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Sign up to join this communityI have the DE
$y''-4y'+4y=e^{2x}$
The general solution to the corresponding homogeneous equation is,
$y_h(x)=e^{2x}(A+Bx)$
How do I find a particular solution $y_p(x)$ that fits with this?
There's a method called "the method of undetermined coefficients" that deals with such equations. Normally, when you see $e^{2x}$ on the right-hand side you try $y_p = Ae^{2x}$, plug it in, then solve for $A$. However, since $2$ is a double root of the characteristic equation $r^2 - 4r + 4 = 0$, you have to try $y_p = Ax^2 e^{2x}$ instead. If you plug this in, and solve for $A$, you obtain $A = {1 \over 2}$ and therefore $y_p = {1 \over 2}x^2 e^{2x}$.
$y^{\prime \prime} -4y^{\prime} + 4y = e^{2x} \quad \Rightarrow \quad (D^2 - 4D + 4)y = e^{2x} \quad \Rightarrow$
$(D -2)^2y = e^{2x} \quad \Rightarrow \quad y = \frac{1}{(D - 2)^2}e^{2x} = e^{2x}\frac{1}{D^2}\cdot 1 = \frac{x^2}{2}e^{2x}$