MATH 20D Lecture 2

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

Section 1.2 - Initial Value Problems

Initial Value Problem (IVP)

An initial value problem (IVP) is a differential ewquation that has its conditions specified at some value .

$y^{\prime }\cdot y=x,y(2)=-1$

Previously, we saw that every function of the form is a solution to
We refer to as general solution.
We now use the initial condition/value: means when .
To solve for C,
Thus, is a particular solution to the given IVP

Verify that any function of the form $y(x)=c_1{e}^x+c_2{e}^{-2x}$ for arbitrary constants $c_1,c_2$ is a solution to the differential equation $\frac{d^{2}y}{d{x}^2}+\frac{dy}{dx}-2y=0$

So we plug them in:

Thus, is general solution to

$\frac{d^{2}y}{d{x}^2}+\frac{dy}{dx}-2y=0,y(0)=2,y^{\prime }(0)=1$

So far, we know are general solutions.
We now use the inital condition to solve for .
When
Recall so when
Thus,
We obtain a system of 2 equations, 2 unknowns

Hence, the particular solution is:

In general, an n-th order initial value problem requires n initial conditions:

Explicit Solution

A function f(x) is called an explicit solution to the given ODE on an interval I provided that, when substituted for y, it satisfies the given equation for all in I.

Implicit Solution

A relation G(x,y)=0 is said to be an implicit solution to the given ODE on the interval I if it defines one or more explicit solutions on I.

General Solution

The general solution is the solution of the given ODE that contains
an abitrary constant C.

Particular Solution

he particular solution is the solution to the IVP, obtained from the
general solution by solving for C using the given initial value(s)

In general an IVP can have

No solution
A unique solution
Infinitely many solutions

Section 1.3 - Direction Field

The logistic equation for the population p (in thousands) at time t of a certain species is given by $\frac{dp}{dt}=p(2-p)$

Equilibrium solutions: 0, 2
p=2: sink/stable equilibrium
p=0: source/unstable equilibrium

Phase line

We need the phase line to study the behavior of the solution .
Vertical line pointing upwards with P at the tip. p=0, p=2 (points at the roots). Examining where the first derivative of the function is positive and negative (when the population is increasing and decreasing). Draw smaller arrows that show which way the population goes in relation to the roots.

Autonomous Differential Equations

Differential equations of the form are called autonomous.
or an autonomous equation, the solutions y(t) = yi, where yi’s are the
roots of f (t), are called equilibrium.

• If an equilibrium repels neighboring solutions as t → ∞ then it is called a source, repeller, or unstable equilibrium,
• If an equilibrium attracts neighboring solutions as t → ∞ then it is called a sink, attractor, or stable equilibrium,
• If an equilibrium is neither a source nor a sink, then it is called a node, or semi-stable.