12.1 Consider the nonlinear first-order equation x = 2√x - 1. (a) By separating solution x₁(1). (b) Your solution should contain one constant of integration k, so you might reasonably expect it to be the general solution. Show, however, that there is another solution, x₂() = 1, that is not of the form of x₁ (1) whatever the value of k. (c) Show that although x₁(t) and x₂(t) are solutions, neither Ax₁(1), nor Bx₂(1), nor x₁(1) + x₂(t) are solutions. (That is, the superposition principle does not apply to this equation.)
12.1 Consider the nonlinear first-order equation x = 2√x - 1. (a) By separating solution x₁(1). (b) Your solution should contain one constant of integration k, so you might reasonably expect it to be the general solution. Show, however, that there is another solution, x₂() = 1, that is not of the form of x₁ (1) whatever the value of k. (c) Show that although x₁(t) and x₂(t) are solutions, neither Ax₁(1), nor Bx₂(1), nor x₁(1) + x₂(t) are solutions. (That is, the superposition principle does not apply to this equation.)
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Transcribed Image Text:12.1 Consider the nonlinear first-order equation * = 2√x-1. (a) By separating variables, find a
solution x₁(1). (b) Your solution should contain one constant of integration k, so you might reasonably
expect it to be the general solution. Show, however, that there is another solution, x₂(t) = 1, that is
not of the form of x₁(1) whatever the value of k. (c) Show that although x₁(t) and x2(t) are solutions,
neither Ax₁(1), nor Bx₂(1), nor x₁(1) + x₂(t) are solutions. (That is, the superposition principle does
not apply to this equation.)
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