Question

Solve (D^2+5D+4)y=x^2+7x+9​

Answer 1

Answer: - The answer to the given equation is

                = Ae^−4x+Be^−x+8x²+36x+23/32

Detailed solution: -

This is the second order Linear differential equation, with the constant co-efficient.

The characteristic equation of this is: -

$m^2+5m+4=0$

Here,

m = -1, -4.

And,

The Characteristic Function for this is,

C.F. = Ae^(-x) + Be^(-4x)

(D² + 5D + 4) y = x² + 7x + 9

$y_{p} = \frac{1}{(D+4)(D+1)}*(x^2+7x+9)\\ \\=\frac{1}{4} (1+D)^-^1(1+\frac{D}{4})^-^1(x^2+7x+9)\\\\ =\frac{1}{4} (1-D+D^2- ...)(1-\frac{D}{4}+(\frac{D^2}{16})-...(x^2+7x+9)$

You need to leave out the terms which are higher than D².

$=\frac{1}{4}(1-\frac{5D}{4}+\frac{21D^2}{16})(x^2+7x+9)\\ \\ =\frac{1}{4}(x^2+7x+9-\frac{5(2x+7)}{4} +\frac{21}{16}*2)\\ \\ =\frac{8x^2+36x+23}{32\\}\\\\ y=y_{c} +y_{p\\} \\\\=Ae^-^4^x+Be^-^x+\frac{8x^2+36x+23}{32}$

Generally,

The general solution to the given equation is,

y = C.F. + P.I.

Therefore,

Your answer to the given equation is = Ae^−4x+Be^−x+8x²+36x+23/32

To know more about the topic, go to the given links: -

https://brainly.in/question/54178540

https://brainly.in/question/36142223

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