SOLUTION: Q.The variables x and y are related by the differential equation (x2 + 4)dy/dx = 6xy. It is given that y = 32 when x = 0. Find an expression for y in terms of x. The answer is

Algebra ->  Complex Numbers Imaginary Numbers Solvers and Lesson -> SOLUTION: Q.The variables x and y are related by the differential equation (x2 + 4)dy/dx = 6xy. It is given that y = 32 when x = 0. Find an expression for y in terms of x. The answer is      Log On


   

Question 749772: Q.The variables x and y are related by the differential equation
(x2 + 4)dy/dx = 6xy.
It is given that y = 32 when x = 0. Find an expression for y in terms of x.
The answer is supposed to be: y=0.5(x^2+4)
but i get a different answer... please help soon exam is one day away!#Desperate
Thank you.
Answer by blueplanet(7) About Me  (Show Source):
You can put this solution on YOUR website!
Here is my solution:
+%28x%5E2+%2B+4%29%2A%28dy%2Fdx%29+=+6%2Ax%2Ay+ ------- (1)

Rearranging the equation, we get:
+%28dy%2Fy%29+=+6%2Ax%2F%28x%5E2%2B4%29%2Adx+ ------- (2)

Replacing x%2Adx with d%28x%5E2%2B4%29%2F2, we get:
+%28dy%2Fy%29+=+3%2F%28x%5E2%2B4%29%2Ad%28x%5E2%2B4%29%29+ ------- (3)
Integrating both sides of the equation, we get:
+int%28dy%2Fy%29+ = int%283%2F%28x%5E2%2B4%29%2Ad%28x%5E2%2B4%29%29+%29+
+ln%28y%29+=+3%2A+ln%28x%5E2%2B4%29+ -------- (4)
This expression can be simplified into:
+y+=+%28x%5E2%2B4%29%5E3+
I think this is the solution to the problem. I have checked the solution by differentiating it with respect to x. The end result is exactly the same as the equation given in the question:
+y+=+%28x%5E2%2B4%29%5E3+
+dy%2Fdx+=+3%2A%28x%5E2%2B4%29%5E2+
+dy%2Fdx+=+3%2A%28x%5E2%2B4%29%5E3%2F%28x%5E2%2B4%29+
Replacing %28x%5E2%2B4%29%5E3 with y, we get:
+dy%2Fdx+=+3%2Ay%2F%28x%5E2%2B4%29+
Rearranging the equation, we get an equation exactly exactly the same as the equation given in the original question:
+%28x%5E2+%2B+4%29%2A%28dy%2Fdx%29+=+6%2Ax%2Ay+