some calculas questions, plz help?

Question

hi every1, i have some questions in calculas and i was hoping to have some answers..

1- if F(x) + F'(x) = X^3 + 5X^2 + 2 (Find F(x))

2- if X^n * Y^m = (x+y)^(m+n), prove that (dy/dx = y/x)

thanks..

Answer 1

F(x) + F'(x) = x^3 +5x^2 + 2

Let F(x) = Ax^3 + Bx^2 + Cx + D
Then F'(x) = 3Ax^2 + 2Bx + C

F(x) + F'(x) = Ax^3 + Bx^2 + Cx + D + 3Ax^2 + 2Bx + C

Group like terms.

F(x) + F'(x) = Ax^3 + Bx^2 + 3Ax^2 + Cx + 2Bx + C + D
F(x) + F'(x) = Ax^3 + (B+3A)x^2 + (C+2B)x + (C + D)

Now, equate each coefficient you solved for here, for the coefficients on the right hand side of the original equation. You'll notice that

A = 1
B+3A = 5
C+2B = 0
C+D = 2

After solving that system of equations, you should end up with
A = 1, B = 2, C = -4, and D = 6

Therefore, substitute the values here appropriately.
F(x) = Ax^3 + Bx^2 + Cx + D
F(x) = x^3 + 2x^2 - 4x + 6

Answer 2

19

Answer 3

1)The last answer is almost there.
The complete solution is:

F(x) = x^3 + 2x^2 - 4x + 6 + Ce^(-x); C constant

After finding the particular solution:
p(x) = x^3 + 2x^2 - 4x + 6
add the kernel:
ie., the function g(x) for which
g(x) + g'(x) = 0
or
g'(x)/g(x) = -1
∫g'(x)/g(x) dx= ∫-1dx
ln(g(x)) = -x + c
g(x) = Ce^(-x)

and F(x) = p(x) + g(x)

This can also been derived with the general solution to the linear differential equation:
y' + y = q(x) →

y = e^(-x) * [C + ∫e^x * q(x) dx]
= Ce^(-x) + e^(-x)*∫e^x * q(x) dx
which means:
y' = -Ce^(-x) + e^(-x)*e^x*q(x) - e^(-x)*∫e^x * q(x) dx
= -Ce^(-x) + q(x) - e^(-x)*∫e^x * q(x) dx

So, y + y' = q(x)