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