hw1 - By separating variables, find the solution of the...

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Unformatted text preview: By separating variables, find the solution of the following initial value problems in explicit form: 1. y ′ = 2 x/ ( y + x 2 y ), y (0) =- 2. 2. y ′ = xy 3 (1 + x 2 ) − 1 / 2 , y (0) = 1. 3. y ′ = 3 x 2 / (3 y 2- 4), y (1) = 0. (Leave solution as a cubic equation for y .) 4. y ′ = (2- e x ) / (3 + 2 y ), y (0) = 0. 5. Find the solution of the following initial value problem with x = x ( t ) and ˙ x = dx/dt : ˙ x + 2 x = te − 2 t , x (1) = 0 . 6. Consider the initial value problem: ˙ x + 2 3 x = 1- 1 2 t, x (0) = x . Find the value of x for which the solution touches, but does not cross, the t-axis. 7. Find the value of x for which the solution of the initial value problem ˙ x- x = 1 + 3 sin t, x (0) = x remains finite as t → ∞ . 8. Consider the initial value problem ˙ x- 3 2 x = 3 t + 2 e t , x (0) = x . Find the value of x that separates solutions that grow positively as t → ∞ from those that grow negatively....
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