MATH 20D Lecture 1

Source: https://canvas.ucsd.edu/courses/52767/files/11229957?module_item_id=2111692
Source: https://canvas.ucsd.edu/courses/52767/files/11240075?module_item_id=2119876

Differential Equation

An equation that contains some derivatives of an unknown function.
If the equations involves the derivatives of one variable with respect to another then the former is called a dependent variable and the latter is called an independent variable.

$\frac{d^{2}y}{d{x}^2}-\frac{2y}{x}=0$

unknown is function
: independent variable
or : dependent variable

Ordinary Differential Equation (ODE)

only one independent variable

Order of a Differential Equation

Order of a differential equation is the order of the highest-order derivative present.

Order of $\frac{dy}{dx}+y^3=\sin (y)$

1st order given by

Linear vs. Nonlinear ODE

an n-th order linear ODE can be written in the form

Otherwise, the ODE is said to be nonlinear

Solution to a Differential Equation

A solution to a differential equation (DE) on an interval is all functions which satisfies the given DE for all values .

The differential equation $y^{\prime }=f(t)$ has the general solution given by $y(t)=F(t)+C$

where is the anti-derivative of and is an arbitrary constant.

The differential equation $x\frac{dy}{dx}=2y$ has a solution $y(x)=x^2$

LHS=RHS

The differential equation $y^{\prime }\cdot y=x$ has an (implicit) solution given by $x^2-y^2=C$ for any arbitrary constant C.

implicit definition needs implicit differentiation