MATH 20D Lecture 3

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.

Equilibrium solution

Occurs at the points where

$\frac{dy}{dx}=(y-1)(y-2{)}^2(y-3)$

Equilibrium solutions:
when
is a source/repeller/unstable
when
is a node/semi-stable
when
is a sink/attractor/stable
when

$\frac{dy}{dx}=y\sin (y)$

when
when
when
when
when
when
when
when

Chapter 2: First-Order ODEs

Three main types of First-Order Equations

Separable ODE

Linear (1st order) ODE

Exact ODE

where

Separable ODEs

$x^2\frac{dy}{dx}+\sin (y)=0$

$xy\frac{dy}{dx}-(y-x{)}^2=0$

Assume that
is inseparable, so this is NOT a separable ODE

$(1-y^2)\frac{dy}{dx}=x^2$

Solving Separable ODEs

“Separate then integrate”

1. Convert the given equation into the form
2. Integrate both sides to obtain . This gives the implicit solution to the given equation. (General solution in implicit form)
3. Convert the solution obtained in step 2 to explicit form and use the initial condition to solve for the constant C, if required/able.

Solve the following IVP: $\frac{dy}{dx}=\frac{3x^2+4x+2}{2(y-1)},y(0)=-1$

1. Separate
2. Integrate
   
3. Use initial condition to solve for C:
   Thus, the particular soln in implicit form is
   To convert this into explicit form:
   
   
   
   
   There can only be 0, 1, or infinitely many solutions. Not 2, so we have to figure out if we are using the + or - sign.
   Under the inital condition when , which is if the sign is +, or if the sign is -
   So the correct sign is -
   is the particular solution in explicit form.

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

1. Separate
2. Integrate