Solve the differential equation.
-$$\frac { d y } { d x } = 8 x ^ { 7 } \sqrt { y - 1 }$$
A) $y = \left( 2 x ^ { 8 } + C \right) ^ { 2 } + 1$
B) $y = \frac { 1 } { 4 } x ^ { 16 } + C$
C) $y = \left( \frac { 1 } { 2 } x ^ { 8 } + C \right) ^ { 2 } + 1$
D) $y = \left( x ^ { 8 } + C \right) ^ { 2 }$

Question

Quizplus · Accepted Answer

By separating variables and integrating, we have:

$$\int \frac{1}{\sqrt{y-1}} dy = \int 8x^7 dx $$

Applying the substitution $u = y - 1$, $du = dy$, we get:

$$2 \sqrt{y-1} = \frac{8}{8} \cdot x^8 + C$$

Solving for $y$, we get:

$$y = \left(\frac{1}{2}x^8 + C\right)^2 + 1$$

Therefore, the answer is (C).

Solve the Differential Equation dydx=8x7y−1\frac { d y } { d x } = 8 x ^ { 7 } \sqrt { y - 1 }dxdy​=8x7y−1​

Solve the Differential Equation $\frac { d y } { d x } = 8 x ^ { 7 } \sqrt { y - 1 }$