1. Sec 1.5: Linear First-Order DE

2. Sec 1.5: Linear First-Order DE
Definition 2.2
A 1st order De of the form
is said to be linear first-order equation. 1 2 3

3. How to Solve ? Step 1 Step 2 Step 3 Step 4 Method of Solution:
Sec 1.5
How to Solve ?
Method of Solution:
Rewrite into standard form
(coeff of y’ is 1).
Step 1
----- (1)
Find the Integrating Factor =
( ignore constant of integration)
Step 2
Step 3
Multiply (1) by ( check: )
Step 4
Integrate both sides:
( DONOT forget constant of integration) 1 2 3

4. Remark: linear in y or linear in x
Sec 2.3
Remark: linear in y or linear in x
Solve the IVP
Find a general solution of
Find a general solution of

5. Derivation
The solution of the above DE is given:
where

6. Theorem 1: Theorem 1: The Linear First-Order Equation Remarks:
If the function and are continuous on the open interval I containing the point , then the initial value problem
Theorem 1 gives a solution on the entire interval I
Theorem 1 tells us every solution is included in the formula (6)
Theorem 1 tells us that the general solution is given in(6)
Theorem 1 tells us that a linear first-order DE has no singular sol
has a unique solution y(x) on I, given by the formula in Eq(6) with an appropriate value of C.
(6)
where

7. Sec 2.3
Solve the IVP 1

8. Derivative and Integration
MATHEMATICA
D[x^3+3 x,{x,2}]
Integrate[ x^2 , x]
syms x
diff(x^3+3 x)
syms x
int(x^2,x)
MATLAB