Differential equation?

Question

Okay guys, I'm stumped. Or maybe I'm too tired to think it through straight...

I have to solve the differential equation:
(dy/dx) = (e^x/y)
Taking into account that y = 2 when x = 0.

I then have to work out y when x = 0.5. Any help would be appreciated.

Answer 1

It looks like a separable diff eq to me. That means you can get everything involving x on one side and y on the other, even the dy and dx:

dy/dx = e^x / y

dy/dx * y = e^x (multiplying both sides by y)

dy * y = e^x dx (informally, think of multiplying both sides by "dx")

now integrate both sides: left side with respect to y because of the "dy", right side with respect to x because of the "dx":

int (y dy) = int (e^x dx)
(1/2)y^2 = e^x + C (only need the +C on one side)

finally, solve for y:

y^2 = 2e^x + 2*C
y = sqrt(2e^x + 2*C)

but what is C?

well when x=0, y=2:

2 = sqrt(2e^0 + 2*C)
2 = sqrt(2 + 2*C)
4 = 2 + 2*C
2 = 2*C
C = 1

so the solution is y = sqrt(2e^x + 2)

check:

dy/dx = e^x/(sqrt(2e^x + 2) = e^x / y

Answer 2

Is the right side exp(x/y) or (exp x) / y? If the latter, then it's easy, for you can separate variables: rewrite (using differentials) as

y dy = exp x dx,

and integrate both sides to get 1/2 y^2 = exp x + c, which can be solved for either variable. You determine c by plugging in the initial value: 1/2 * 4 = 1 + c, whence c = 1. Solution is

1/2 y^2 = 1 + exp x.

Now, if it's exp(x/y), that's quite a bit harder. It's nonlinear and nonautonomous. I do not know of a solution method in this case. There is a graphical solution available, however.

Answer 3

If this is

dy/dx = e^x / y then it is of the type variable seperable

collecting x terms on one side and y terms on the other side

ydy = e^x dx

integrating

∫y dy = ∫ e^x dx

½ y² = e^x + c is the general solution

y = 2 , x = 0

2 = 1 + c

c = 1

½ y² = e^x + 1 is the particular solution

when x = 0.5

½ y² = e^0.5 + 1

y² = 2(e^0.5 + 1) = 5.29744...

y = 2.301(6.....)

Answer 4

dy/dx = (x^2)(8 + y) First, we separate the variables by potential of multiplying by potential of dx and dividing by potential of (8 + y): dy/(8 + y) = (x^2)dx combine the two sides, remembering the left will use a organic log and the wonderful could have a persevering with: ln(8 + y) = (a million/3)x^3 + C improve e to the flexibility of the two part: 8 + y = Ae^((a million/3)x^3) Subtract 8 and we've the ordinary variety: y = Ae^((a million/3)x^3) - 8 Now, we've the element (0, 3), so plug those in to locate a and the specific answer: 3 = Ae^((a million/3)(0)^3) - 8 remedy for A: 11 = Ae^((a million/3)(0)) 11 = Ae^(0) A = 11 So, the specific answer is: y = 11e^((a million/3)x^3) - 8

Answer 5

dy/dx = e^x/y
y dy = e^x dx 'integrate both sides
(1/2)y^2 = e^x + C 'use (0,2) to find C
(1/2)2^2 = e^0 + C
C = 1
Solution to the differential : y^2 = 2e^x + 2
Plug in x = .5 and find y.

Answer 6

∫ y dy = ∫ e^x dx
y² / 2 = e^x + c
2 = 1 + c
c = 1
y² / 2 = e^x + 1
y² / 2 = e^0.5 + 1
y² / 2 = 1.659 + 1
y² / 2 = 2.659
y² = 5.318
y = 2.306

Answer 7

well when somethnig is to a power of 0 i know that it is equal to one.So I think your equation would become (d2/d0)=(1/2).Then you would have to go from there.Sorry I can't help more.I think everything I said is correct.

Answer 8

well, the answers are so good. i know differential very well but i do not need to answer this question. thanks