MATH 20D Lecture 5

$a{\log }_b(x)={\log }_b(x^a)$

Solve the IVP $x{y}^{\prime }+2y=x^2+x-1,y(1)=-1/2$

Convert into standard form: (assume )
Here so
So
The integrating factor is
Now multiply to both sides of the standard form eqn





Now we use initial value

Therefore, particular soln in explicit form is

Why does the choice of $u(x)=e^{\int P(x)dx}$ work?

Recall the standard form of our 1st-order linear ODE:
After multiplying to both sides of this standard form:

So to turn LHS into a single derivative w/ product rule, we want
So is a soln to the ODE:
This is a separable equation

Exact ODEs

An exact equation has the form

where “cross deriv.”

Method for Solving Exact Equations

In general, if is exact then there exists a function such that

The *implicit solution is then given by *

Solve $(y^4-2)+4x{y}^3\frac{dy}{dx}=0$

1. Integrate with respect to to obtain
   
   
2. Integrate with respect to to obtain
   
   
3. Match the annswers from the previous steps to determine
   
   
   Implicit soln: or