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Alfy102
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Homework Statement
If y(1+x2) dy/dx = 2x (1-y2), prove that (1+x2)2(1-y2)=A, where A is constant.
Homework Equations
Separable equations
The Attempt at a Solution
Separate the terms:
y/(1-y2) dy = 2x/(1+x2) dx
Integrating both sides will get:
∫ y/(1-y2) dy = ∫ 2x/(1+x2) dx
Use substitution method for ∫ y/(1-y2) dy:
u = 1-y2
du = -2y dy
-du/2 = y dy
∫ -u/2 du = -1/2 ∫ u du
= (-1/2)*(u2/2)
= -u2/4 + C
= -(1-y2)2/4
Use substitution method for ∫ 2x/(1+x2) dx:
u= 1+x2
du = 2x
∫ 1/u du = ln u + C
= ln (1+x2)
Putting them back together will get:
-(1-y2)2/4 = ln (1+x2)
I'm pretty much unable to continue from here.