MATH 20D Lecture 6

Exact Equation

where “cross derivative” or “mixed partial”
If equation is exact, then soln will always be given by where c is some constant
Where

$(3x-e^x\cos y)y^{\prime }=e^x\sin y-3y$

Check for exact:


We look for the general soln of the form



General solution is

$\left(y-\frac{1}{x}\right)y^{\prime }=x-\frac{y}{x^2},y(1)=2$

Check for exact:


We look for the general solution of the form



We plug in and solve for :
Particular solution in implicit form:

$\left(\sin y-\frac{x}{y}\right)dy=dx$

Check for exact:


So this is not an exact equation.
”Clever trick” multiply both sides by y to turn it into an exact equation.

Check for exact:

Method for finding special integrating factor

If the equation is not separable, linear, or exact, then check the following:

1. is a function of only x
2. is a function of only y
   If (1) holds, then the integrating factor is
   
   If (2) holds, then the integrating factor is
   

Chapter 4 - Linear Second-Order ODEs

Second-order linear ODE

with
In our case, we consider only second-order ODE with constant coefficients:

In the special case where , we obtain the homogenous form

For the IVP, we need two initial conditions: and

How to solve homogenous linear second-order ODE

Key idea: look for solutions of the form
Plug in
So
If then
Thus, if r is a solution to (if r is a root of the polynomial ) then will be a soln to the ODE

: has 2 distinct (real-valued) solns. The ODE will have general soln:
: has a repeated soln:
: has complex solutions in the form (conjugate pairs)
is called the characteristic/auxiliary equation

$y^{\prime \prime }+y^{\prime }-2y=0,y(0)=0,y^{\prime }(0)=-3$

Characteristic equation:
General soln is:
Now we use the initial condition to find


So, by solving the system we have .
Thus, the particular soln is