S2

Lecture 2: Differential Equations

1. Classification of Differential Equations

• Order of an ODE (Ordinary Differential Equation):

  • The order is defined as the largest number of times the dependent variable is differentiated in the ODE.

  • Example:

    • The equation ( dy = xy^2 ) is a 1st order ODE.

    • The equation ( \frac{d^2z}{dz^2} = u ) is a 2nd order ODE.

• Linear ODE:

  • An ODE is linear if it contains no products or powers of the dependent variable or its derivatives.

  • Prohibited Terms:

    • Terms like ( y^2, y^3, yy', \sqrt{y}, e^y, \sin(y) ) are not allowed.

  • General Form:

    • ( \frac{d^3y}{dt^3} + a(t) \frac{dy}{dt} + b(t)y = b(t) )

2. Nonlinear and Autonomous ODEs

• Nonlinear ODE:

  • An ODE that is not linear.

• Autonomous ODE:

  • An ODE is autonomous if there is no explicit dependence on the independent variable.

  • Nonautonomous ODE:

    • An ODE that does have explicit dependence on the independent variable.

3. Types of ODEs: Exercises

• Identify the type of the following ODEs by order, linearly, and autonomy:

  • (i) ( y' = 2x^3 )

    • 1st order, linear, nonautonomous

  • (ii) ( ay'' - 7y' + 12y = 5\cos(x) )

    • 2nd order, linear, nonautonomous

  • (iii) ( y'' + 2x^2(y')^2 = 0 )

    • 2nd order, nonlinear, nonautonomous

  • (iv) ( yy''' = (y)^2 )

    • 3rd order, nonlinear, autonomous

  • (v) ( (y')^2 + 2 = e^x )

    • 1st order, nonlinear, nonautonomous

  • (vi) ( (y'')^2 + \sqrt{y} = 1 )

    • 2nd order, nonlinear, nonautonomous

4. General Solution

• General Solution of an ODE:

  • Observing that the most basic solution of the ODE ( \frac{dy}{dt} = 0 ) is the constant function ( y(t) = C_1 ).

• To solve an ODE of n-th order, we integrate n times, leading to a general solution that depends on arbitrary constants ( C_1, ..., C_n.

Example of General Solution:

• Show that ( y(t) = 1 + C ) is a general solution of ( ty^2 + 2y = 2 ):

  • Substituting into the equation explains the derivation of the general solution.

5. Initial Value Problems (IVP)

• Defining IVP:

  • We can fix the values of arbitrary constants in the general solution by imposing extra conditions at a single point ( x_0 ).

  • For 1st order ODE:

    • Structure: ( y'(x) = f(x, y(x)) ) and ( y(x_0) = y_0 )

  • For 2nd order ODE:

    • Structure: ( y''(x) = f(x, y(x), y'(x)) ) and ( y(x_0) = y_0, y'(x_0) = y_1 )

  • Example:

    • For the equation ( y'' + 2y = e ), the IVP is defined with specific initial conditions: ( y(0) = 1, y'(0) = 2 ).

6. Boundary Value Problems (BVP)

• Defining BVP:

  • An alternative for fixing values of arbitrary constants by specifying values of ( y(x) ) and its derivatives at at least two values of ( x ), say ( x_1 ) and ( x_2 ).

  • More complicated than IVPs and not covered in this module.