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.