Question

Use implicit diff. to find dy/dx of each of the following. In the following x,y and (a) are all variables. Show step by step please! Thank you!

1) y^2 = x^2+a^2

2) y^2+ay = x^2+ax+a^2

Answer 1

2 y dy/dx = 2 x + 2 a da/dx

y dy/dx = (x+a da/dx)

dy/dx = (x + a da/dx)/y
or
dy/dx = (x + a da/dx)/(x^2+a^2)^.5
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2 y dy/dx + a dy/dx + y da/dx = 2 x^2 + a + x da/dx + 2 a da/dx

(2y+a)dy/dx = 2 x^2 +a +(x-y+2a)da/dx

dy/dx = [2 x^2 +a +(x-y+2a)da/dx]/(2y+a)

Answer 2

I suspect in each, a is a constant, so da/dx=0

Answer 3

I figured the same but he said specifically that a was variable.

Answer 4

My answers were like these

1)
2yy'=2x+2a

y'=2x+2a/2y

2)
2yy'+ay'+y=2x+a+x+2a-y

2yy'+ay'=3x+3a-y

y'(2y+a)=3x+3a-y

y'=3x+3a-y/2y+a

And my teacher said I made mistakes with my answers but I couldn't figure out.