Differential Equations

EXTRA CREDIT PROBLEMS

1. Let y = f(x), x > 0, be a curve which satisfies the following condition: at each point P on it, the tangent line at P intersects the x-axis at a point Q such that dist(P, Q) = 1. If f(0) = 1, find an equation of the curve (it could be implicit).

2. Find the general solution of 1 y

= y ′′

1+(y′)2 .

3. Find the general solution of y′ = x + x 3

y .

4. Suppose that f(x) satisfies the following equation:

f′ = √ x2 + f2

and f(1) > 0. Show that the graph y = f(x) is concave up for x > 1.

5. Let f : (1,∞) → R be a function such that

f′(x) = x2 − (f(x))2

x2((f(x))2 + 1)

for x > 1. Prove that limx→∞ f(x) = ∞.

6. Find the derivative ∂y ∂µ

at µ = 0 if y = y(x,µ) is a solution of the initial value problem

dy

dx = µx +

1

2y , y(1) = 1 − 2µ, x > 0.

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