Question

Solve the differential equation y prime equals the product of 2 times x and the square root of the quantity 1 minus y squared .

Answer 1

Answer:

1. y = sin(x²+C)

2. see below

Step-by-step explanation:

1. $\frac{dy}{dx} = 2x\sqrt{1-y^2}$

separation of variables

$\frac{dy}{\sqrt{1-y^2} } = 2xdx$

integration both sides$\int\frac{dy}{\sqrt{1-y^2} } = \int2xdx$

you should get :

$sin^{-1} (y) = x^2+C$ remember constant of integration!!

2. y = sin(x²+C)

3 = sin(0+C)

y(0) = 3 does not have a solution because our sin graph is not shifted vertically or multiplied by a factor whose absolute value is greater than 1, so our range is [-1,1] and 3 is not part of this range