Question

How do I find a Particular Integral(PI), for this equation? : \frac{ d^2y }{ dx^2 } + 16y = x^2 + 3

Answer 1

I already have the Complementary Function(CF) it is : \[y _{c} = Acos4x + Bsin4x\]

Answer 2

the equation is actually: \[\frac{ d^2y }{ dx^2 } + 16y = x^2 + 3\]

Answer 3

ok if u have to find paarticular solution then see followung steps

Answer 4

You just let the constant C become a variable。

Answer 5

\[Yp=x ^{2} +3/D ^{2}+16 \]

\[Yp=(x ^{2} +3)(D ^{2}+16)^{-1} \]


\[(D ^{2}+16)^{-1}=D ^{2}-16D ^{4}+..................\]

Answer 6

the next terms including second term can be neglected because derivative higher than third of x is 0

Answer 7

k let me try that now

Answer 8

Yp=\[(x ^{2}+3)*D ^{2}\]
\[D ^{2}x ^{2}+D ^{2}3=2+0\]
Yp=2

Answer 9

This is a nonhomogeneous diffence equation.First of all, you sove the homogeneous equation。

Answer 10

\[\frac{ d^2y }{dx^2}+16y=0\]

Answer 11

homogeneous eqn has been solved and I hv the CF I posted it on the 1st line in this timeline

Answer 12

the Auxillary eqn, had complex roots

Answer 13

Find the eigenequation.

Answer 14

fazeelayza.He actually use the infnit series.Both way are ok.

Answer 15

So why not use the real part of the complex number.

Answer 16

Or he use the Opeator.Ok ,I don't know.

Answer 17

\[\alpha = 0,\beta = 4\], since k = 4i

Answer 18

\[ax^2+bx+c\]plug in the equation to get a b c