Differential equation: solve (y2 + 8) dx = y sec2 x dy

[kheath39]

Solve the given differential equation. (Use C for the constant of integration.)
(y² + 8) dx = y sec² x dy

I'm separating this to get y/(y²+8) dy = 1/sec²x dx which is ln((y²+8)/2 = (sinx cosx + x)/2

I then get y²+8 = e(sinx cosx +x)

y = sqrt(e(sinx cosx + x)-8 + C

Where am I going wrong?

 

[Deleted member 4993]

Solve the given differential equation. (Use C for the constant of integration.)
(y² + 8) dx = y sec² x dy

I'm separating this to get y/(y²+8) dy = 1/sec²x dx which is ln((y²+8)/2 = (sinx cosx + x)/2 ← this where I would apply constant of integration

ln((y²+8)/2 + ln(C₁) = (sinx cosx + x)/2

ln[C(y²+8)]/2 = (sinx cosx + x)/2

and continue.....

I then get y²+8 = e(sinx cosx +x)

y = sqrt(e(sinx cosx + x)-8 + C

Where am I going wrong?

.

 

[Ishuda]

Solve the given differential equation. (Use C for the constant of integration.)
(y² + 8) dx = y sec² x dy

I'm separating this to get y/(y²+8) dy = 1/sec²x dx which is ln((y²+8)/2 = (sinx cosx + x)/2

I then get y²+8 = e(sinx cosx +x)

y = sqrt(e(sinx cosx + x)-8 + C

Where am I going wrong?

As Subhotosh Khan points out, you have to put in the constant of integration at the proper place, i.e.
(ln(y²+8))/2 + C₁= (sinx cosx + x)/2 + C₂
and then consolidate as much as possible at some point. Note that the consolidation Subhotosh Khan gives,
ln[C(y²+8)] = (sinx cosx + x),
where C=e^2(C₁-C₂) provides a different form than that for the consolidation of the constants C=2 (C₂-C₁),
ln[y² + 8] = sin(x) cos(x) + x + C.
However the answer is 'the same' and the two C are related to one another as can be seen.

EDIT: Correct mistake in formula.