# Differential Equations

The is a course on ordinary differential equations. The course will cover integration techniques of first and second order differential equations, linear differential equations and the Laplace Transform. According to time constraint, elementary Sturm-Liouville theory will be introduced. The text book is "Advanced Engineering Mathematics" of Zill and Wright.

### Week Ten

Linear differential equations II

1. Solutions to the equation $(D - \alpha)^2 y + \beta^2 y$ with $\beta\neq 0$. Non-homogeneous equations;
2. Given two polynomials with no common divisor other than constants, $p(D)q(D) y = 0$ if and only if there are $y_1$ and $y_2$ such that $y = y_1 + y_2$
3. non-homogeneous linear equations $Ly = g$. The case where $Lg = 0$.
Exercises: 2, 4, 6, 8, 10, 18, 26, page 127 of the book and the equation $(D^2 + 1)^2 y = 0$. Here you can find the solutions.

### Week Nine

Linear differential equations. Here you can find my notes.

1. Homogeneous equations;
2. solution to a second order linear d.e. with constant coefficients and factorizable characteristic polynomial;
3. solutions to the differential equation $y'' + y = 0$;
4. solutions to the equation $y'' + \beta^2 y = 0$ with $\beta\neq 0$.

### Week Eight (Midterm Exam)

In 10/21 and 10/27 you can find the solutions of the exercises of the Midterm Exam of October 21st and October 27th.

### Week Seven

On this week I showed the solutions of the assignments of Week Four (sol.) and Six (sol.).

### Week Six

Equations which can be solved with a substitution:

1. Bernouilli equations: $y'(x) + P(x)y(x) + Q(x)y(x)^\alpha$ with $\alpha\neq 0,1$;
2. homogeneous equations;
3. equations which can be written as $y'(x) = f(Ax + By + C)$, with $B\neq 0$;
4. second order differential equations $F(x,y'(x),y''(x)) = 0$.

Please, do exercises 4, 6, 8 and 10 (page 45), 8, 10 and 12 (page 53), 36 (page 60), 4, 6, 16, 18, 26, 28 (page 64), 4 (page 142).

### Week Five

Linear equations. Solutions of differential equations which are obtained by pasting two solutions on adjacent intervals. Exact equations, integrating factors.

Please check Section 2.2, for the linear equations, Section 2.4 for the exact equations and these notes.

These are some homeworks. The due date is 2014, October 14.

### Week Four

Locally Lipschitz functions, two variables Lipschitz functions, two variables functions which are Lipschitz on one component. The Picard-Lindelöf Theorem.

Here you can find the notes of my lectures on Lipschitz functions. These are some homeworks. The due date is 2014, October 7.

### Week Three

The Implicit Function Theorem. Lipschitz functions on one variable.

### Week One

Separable differential equations. You can check Section 2.1 of the book and these notes.