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

Question

Quizplus · Accepted Answer

To solve the given differential equation, we can use separation of variables. Rearrange the equation to isolate $y$ terms on one side and $x$ terms on the other. The solution involves integrating both sides, leading to an inverse sine function due to the square root of $1 - y^2/4$ form, which suggests a trigonometric substitution. The correct solution involves recognizing that $y = 2\sin(2x^2 + C)$ satisfies the differential equation, where $C$ is the integration constant. This is because differentiating $y = 2\sin(2x^2 + C)$ with respect to $x$ using the chain rule gives the original differential equation.

Solve the Differential Equation dydx=4x4−y2\frac { d y } { d x } = 4 x \sqrt { 4 - y ^ { 2 } }dxdy​=4x4−y2​

Solve the Differential Equation $\frac { d y } { d x } = 4 x \sqrt { 4 - y ^ { 2 } }$