Unit 4: Differential Equations

What is a homogeneous ordinary differential equation?

Equation in the form dy/dx = F(y/x)

What is the degree of an ordinary differential equation?

The power to which the highest order derivative is raised

How do you check a linear first order ordinary differential equation is exact? {written as Q(x)dy/dx + P(x)y = H(x)}

If P(x) = dQ/dx then the equation is exact

What is the order of an ordinary differential equation?

The highest derivative in the equation

When is an ordinary differential equation nonlinear?

When at least one of the y, dy/dx d²y/dx² etc. are multiplied by each other

When is an ordinary differential equation linear?

When none of the y, dy/dx, d²y/dx² etc. are multiplied by each other

How do you solve a linear exact ordinary differential equation? {written as Q(x)dy/dx + P(x)y = H(x)}

~ can be written as d/dx(Q(x)y) = H(x) ~ ∴ y(x) = [∫H(x).dx + C]/Q(x)

How do you solve a homogeneous ordinary differential equation?

~ use substitution of y = vx ~ differentiate the substitution to get dy/dx = v + x.dv/dx ~ substitute these into original equation to get in terms of v and x and will be a separable equation

What is the integrating factor method of solving differential equations?

~ write in form: dy/dx + Py = Q ~ calculate integrating factor(R): R = e^{∫P.dx} ~ multiply out to find Ry = ∫RQ.dx