Find the general solution for the differential equation. Leave your solution in implicit form. dx/dt=(2-x)sqrt(1-x)

dx/dt = (2-x)√(1-x)

dx/(2-x)√(1-x) = dt

Note that 2-x = 1 + 1-x = 1 + √(1-x)^2

2 arctan √(1-x) = t + c
√(1-x) = tan(t/2 + c) = tan(t/2) + c
1-x = (tan(t/2) + c)^2
x = 1 - (tan(t/2)+c)^2

To find the general solution to the given differential equation, we will follow these steps:

Step 1: Separate the variables.
We begin by rearranging the equation to have the x terms on one side and the t terms on the other side.

dx/sqrt(1-x) = (2-x)dt

Step 2: Integrate both sides.
Next, we integrate both sides of the equation with respect to their respective variables.

∫ dx/sqrt(1-x) = ∫ (2-x)dt

The integral of dx/sqrt(1-x) can be evaluated using the substitution method or the trigonometric substitution method. However, for this particular integral, it can be simplified by using a substitution of u = 1-x:

du = -dx

Substituting du for -dx in the integral, we have:

-∫ du/sqrt(u) = ∫ (2-x)dt

-2∫ du/sqrt(u) = ∫ (2-x)dt

-2∫ u^(-1/2) du = ∫ 2dt - ∫ x dt

Step 3: Evaluate the integrals.
Integrating both sides, we get:

-2 * 2 √u = 2t - (1/2)xt^2 + C

Simplifying further:

-4√u = 2t - (1/2)xt^2 + C

Step 4: Substitute back the original variables.
Since we used the substitution u = 1-x, we need to substitute back x for (1 - u) and simplify.

-4√(1 - x) = 2t − (1/2)x * t^2 + C

This is the general solution to the given differential equation in implicit form.