Differential eqautions

Exam 1 - November 21, 2014

1. (10 points) Find the orthogonal trajectory family of the curve family (γc)c∈R, defined by

γc : R → R2 : x 7→ (x,cx2),

for every c ∈ R. Note that the trace of the curve γc is the parabola

γc(R) = {(x,y) ∈ R2 : y = cx2}.

2. (10 points) Find the differentiable function f : R → R such that:

a. the point (1, 2) belongs to the graph of f;

b. the following relation is fulfilled

f ′(x) = e2x+f(x),

for every x ∈ R.

(That is a Cauchy problem with initial condition f(1) = 2).

3. (10 points) Solve the following Cauchy problem:{ xy′ = −y + x2 + 1

y(1) = 0

4. (10 points) Solve the following differential equation:

x2 −y2 + xyy′ = 0.

5. (10 points) Solve the following differential equation:

y′ = 5y + e−2xy−2.

6. (10 points) Solve the initial-value problem

y′ + y cot x = y3 sin3 x, y(π/2) = 1.

7. (10 points) Use the substitution y = 2x + v − 3 to find the general solution to the equation

y′ = 2 + √ y − 2x + 3.

8. (10 points) Find the orthogonal family to the family of circles

x2 + y2 = r2.

9. (10 points) Solve ln(y)y′ −x2 + 1 = 0.

10. (10 points) Solve ey−x = y′.