Solve the differential equation.
-$$\frac { d y } { d x } = 8 x ^ { 7 } \sec y$$
A) $y = \sin ^ { - 1 } \left( x ^ { 8 } + C \right)$
B) $y = \sin \left( x ^ { 8 } + C \right)$
C) $y = \cos ^ { - 1 } \left( x ^ { 8 } + C \right)$
D) $y = x ^ { 8 } + C$

Question

Quizplus · Accepted Answer

We have $\frac { d y } { d x } = 8 x ^ { 7 } \sec y$. Integrating both sides with respect to $x$ we get $\int\frac {1}{\sec y}dy =  \int 8 x ^ { 7 } dx$. The left-hand side can be simplified using the identity $\sec^2 y=1+	an^2y$. Thus, we have $\ln| \sec y + 	an y | = x^8+C$ where $C$ is the constant of integration. Taking the inverse sine of both sides, we get $\sin^{-1}(\sec y + 	an y) = x^8+C$, which can be rearranged to obtain $y = \sin^{-1}(x^8+C)$. Therefore, the best choice is $\boxed{	ext{(A) } y = \sin ^ { - 1 } \left( x ^ { 8 } + C ight)}$.

Solve the Differential Equation dydx=8x7sec⁡y\frac { d y } { d x } = 8 x ^ { 7 } \sec ydxdy​=8x7secy

Solve the Differential Equation $\frac { d y } { d x } = 8 x ^ { 7 } \sec y$