Solve: d^2y / dx^2 + 4 dy/ dx + 4y = 0?

Question

Hint: Second order linear equation..
Comment: Not a great problem!!

Answer 1

(D^2+4D+4)=0
auxillary eq is
m^2+4m+4=0
(m+2)^2=0
m=-2,-2
complementary func.,CF=(Ax+B)e^-2x
particular integral,PI=(1/D^2+4D+4)*0 =0
compleate sol. is
y=CF+PI=(Ax+B)e^-2x +0

Answer 2

First, find the roots of the auxiliary equation, which in this case is a quadratic equation:

u^2 + 4u + 4 = 0

Both roots are -2. Thus the solution of the differential equation is

y = A exp(-2x) + B x exp(-2x)

Let's check...

dy/dx =
-2 A exp(-2x) + B exp(-2x) - 2 B x exp(-2x)

d^2y/dx^2 =
+4 A exp(-2x) - 4 B exp(-2x) + 4 B x exp(-2x)

The differential equation is
d^2y/dx^2 + 4 dy/dx + 4y = 0

Substituting the expressions for the derivatives gives
+ 4 A exp(-2x) - 4 B exp(-2x) + 4 B x exp(-2x)
-8 A exp(-2x) + 4 B exp(-2x) - 8 B x exp(-2x)
+4 A exp(-2x) + 4 B x exp(-2x)

All terms cancel. The sum is zero. The solution checks.

Answer 3

The equation is y" + 4y' +4y = 8 sin 2x. Given the particular answer, we could p.c. p and q such that the particular answer satisfies the diff. eq. we detect y = p cos 2x + q sin 2x, y' = -2p sin 2x + 2q cos 2x, and y" = -4p cos 2x - 4q sin 2x. for that reason. y" + 4y' + 4y = -8p sin 2x + 8q cos 2x. that's = 8 sin 2x whilst p = -a million and q = 0, so we've the particular answer (a million) y_p = - cos 2x. For the known answer of the homogeneous equation, we've the auxilliary equation u^2 + 4u + 4 = 0, which has 2 equivalent roots u = -2. This tells us that we've the known answer y = Ae^(-2x) + Bxe^(-2x). we want this to be a million whilst x = 0, so we detect a = a million, and our answer is y = e^(-2x) + Bxe^(-2x). on the grounds that this could be 0 whilst x = 0, we see that B = 0. this provides us the answer (2) y_g = e^(-2x). Our finished answer is y_p + y_g.

Answer 4

z^2 + 4z + 4 = 0 gives z = -2 as a double root.
So y = (a + bx)*exp(-2x) where a and b are constants.
There's nothing to think about here. Just look at the rules in your DFQ book.

Answer 5

Try to substitute a for dy/dx. Then solve the quadratic.

Answer 6

Wow! Thats a nice question, anybody, got Shakuntala's no.??

Answer 7

D^2+4D+4y = 0
D(D+4) = -4y

Thus .............

Answer 8

Shakuntala is a moron, just so you know.