# Find G. y" - 4y' + 3y = 3x^2 - 2x - 3. H. y" - 4y' + 3y = 9x^2 I. y' - 4y = 2e^2x J. y' -4y =...

## Question:

Find

G. {eq}y" - 4y' + 3y = 3x^2 - 2x - 3{/eq}

H. {eq}y" - 4y' + 3y = 9x^2{/eq}

I. {eq}y' - 4y = 2e^{2x}{/eq}

J.{eq}y' -4y = 2e^{3x}{/eq}

Q. {eq}y'- 4y = (- 8x^2 + 2x - 4)e^x{/eq}

R. {eq}y' - 4y = x^2e^x{/eq}

S. {eq}y' - 4y =(- 8x - 4)e^{4x}{/eq}

## Non-Homogeneous Second Order Differential Equation:

To solve a non-homogeneous linear differential equation of second order with constant coefficients,

{eq}a\frac {\partial^2 y}{\partial x^2} +b\frac {\partial y}{\partial x} +cy =g(x)\\ a \,\,\, b \,\,\, c \,\,\,\, \textrm {constants}\\ {/eq}

we must look for the complementary solution {eq}y_c {/eq} and find the particular solution {eq}y_p {/eq} of the non-homogeneous equation.

Then, the general solution is: {eq}y = y_c + y_p {/eq}

One of the methods to find the particular solution is that of undetermined coefficients. This method is used when g (x) has a special form, involving only polynomials, exponentials, sines and cosines, or g (x) is a finite linear combination of this functions.

The method is called the undetermined coefficients method because it is a coherent proposal about the form of the particular solution.

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Find G. {eq}\displaystyle y" - 4y' + 3y = 3x^2 - 2x - 3 {/eq}

Step 1. Calculate the solution of the associated homogeneous equation

{eq}\disp... 