1. First-Order Differential Equations Part 1

2. First-Order Differential Equations
Types:
Variable Separable
Linear Equations
Exact Equations
Solvable by Substitutions

3. Variable Separable
The simplest of all differential equations are those of the first order with separable variables.
A first-order differential equation of the form
is said to be separable or to have separable variables.

4. H(y) and G(x) are antiderivatives of p(y) and g(x), respectively.
Variable Separable
To solve variable separable first-order differential equations, proceed as follows:
Let 1/h(y) = p(y)
H(y) and G(x) are antiderivatives of p(y) and g(x), respectively.

5. Example
Solve:
Solution:

6. Alternate Solution:

7. Example
Solve:
Solution:

8. Example
Initial Boundary Condition:
Solving for C:

9. Example
Initial Boundary Condition:
Solving for C:

10. Example
Alternate Form of Final Answer:

11. Linear Equations A first-order differential equation of the form
is said to be a linear equation in the dependent variable y.
When g(x) = 0, the linear equation is said to be homogeneous; otherwise, it is nonhomogeneous.

12. Linear Equations
We can divide both sides of the equation by the lead coefficient a1(x):

13. Linear Equations Standard form of a linear 1st-order DE:
This differential equation has the property that its solution is the sum of two solutions:
y = yc + yp

14. Linear Equations
Now, yc is a solution of the associated homogeneous equation
and yp is a particular solution of the nonhomogeneous equation:

15. Linear Equations
Proof:

16. Now, the previous homogeneous equation is also separable:

17. Now, let yp = u(x)y1:

18. Separating variables and integrating gives

19. Now, going back to yp = uy1:
Now, going back to y = yc + yp:

20. Remember this special term called the “integrating factor”:
We can use the integrating factor as follows:
General Solution

21. Standard form of a linear 1st-order DE:
RECALL AGAIN
Standard form of a linear 1st-order DE:
Left-hand side of the standard form, to be used for deriving the solution (see next slide)

22. Left side of standard form of 1st order, linear DE multiplied by .
A Simpler Derivation
This derivation hinges on the fact that the left hand side of the 1st-order differential equation (in standard form) can be recast into the form of the exact derivative of a product by multiplying both sides of the equation by a special function (x).
Left side of standard form of 1st order, linear DE multiplied by .

23. Derivative of a product of two variables
Left-hand side of the standard form of a 1st order linear D.E. ?

24. A Simpler Derivation
The derivation then involves solving the encircled elements as follows:

25. A Simpler Derivation
Continuing, we have:

26. A Simpler Derivation
Even though there are infinite choices of (x), all produce the same result. Hence, to simply, we let c2 = 1 and obtain the integrating factor.

27. A Simpler Derivation
This is what we have derived so far. We multiply both sides of the standard form of the 1st-order equation by the integrating factor (x).
We can then integrate both sides of the resulting equation and solve for y, resulting in a one-parameter family of solutions.

28. A Simpler Derivation

29. Solving a Linear, 1st-Order DE
Put the differential equation in standard form.
From the standard from, identify P(x) and then find the integrating factor
Multiply the standard form equation by the integrating factor.

30. Solving a Linear, 1st-Order DE
The left hand side of the resulting equation is automatically the derivative of the integrating factor and y:
Integrate both sides of this last equation.

31. Example
Find the general solution of:
Solution:
Step 1:

32. P(x): include the negative sign if present
Example
Solution:
Step 2:
Integrating Factor:
P(x): include the negative sign if present

33. Example
Solution:
Step 3:
Step 4:
Recall:

34. Example Solution: Step 3: Step 4: Thus: Derivative of y
Derivative of x –3

35. Example
Solution:
Step 5:

36. Exercises
Solve the following variable separable differential equations. 1. 2. 3. 4. 5.

37. Answers:

38. Exercises
Solve the following 1st-order, linear differential equations. 1. 2. 3. 4. 5.

39. Answers: