Solve the differential equation.
-$$\frac { d y } { d x } = 6 x ^ { 5 } \cos ^ { 2 } y$$
A) $y = x ^ { 6 } + C$
B) $y = \tan ^ { - 1 } \left( x ^ { 5 } + C \right)$
C) $y = \tan \left( x ^ { 6 } + C \right)$
D) $y = \tan ^ { - 1 } \left( x ^ { 6 } + C \right)$

Question

Quizplus · Accepted Answer

We can separate the variables and integrate both sides:
$$\int \frac { d y }{\cos ^ { 2 } y} = \int 6 x ^ { 5 } d x$$
Using the substitution $u = \sin y$ and $du = \cos y$:
$$\tan y = - \frac { 1 } { 6 } x ^ { 6 } + C$$
Taking the inverse tangent of both sides gives us:
$$y = \tan ^ { - 1 } \left( - \frac { 1 } { 6 } x ^ { 6 } + C \right)$$
which matches choice D.

Solve the Differential Equation dydx=6x5cos⁡2y\frac { d y } { d x } = 6 x ^ { 5 } \cos ^ { 2 } ydxdy​=6x5cos2y

Solve the Differential Equation $\frac { d y } { d x } = 6 x ^ { 5 } \cos ^ { 2 } y$