1. Families of Solutions, Geometric Interpretation
MATH 374 Lecture 2
Families of Solutions, Geometric Interpretation

2. 1.3: Families of Solutions
Consider the following differential equations:
To solve (1), we can integrate (hopefully), and find
where C is any real number.

3. One-Parameter Families
In fact, since all antiderivatives of f(x) on an interval differ by a constant, all solutions of (1) are given by (3).
We say that (1) has a one-parameter family of solutions.

4. One-Parameter Families
It turns out that all solutions of (2) are given by the one-parameter family:
y = C ek x, (4)
where C is any real number.
In general, what we find is that solutions to an nth order ODE (if they can be found) will occur as an n-parameter family of solutions.
To find a particular solution to a given differential equation, we will need to have more information.

5. Example 1
Find the solution to
that passes through (x,y) = (1,5).

6. Example 2 We will find that all solutions to y’’ + y = 0 (6)
are of the form
y = c1 cos x + c2 sin x. (7)
To find a particular solution to (6), we need to specify two pieces of information, such as
y(0) = 1 and y’(0) = -4. (8)

7. Example 2 Putting (8) into (7), we can find c1 and c2:
1 = y(0) = c1 cos(0) +c2 sin(0) = c1
-4 = y’(0) = -c1 sin(0) + c2 cos(0) = c2
Therefore,
y = cos x – 4 sin x.

8. Elimination of Constants
Given a family of curves with parameters, we can often find an associated differential equation.
The way to do this is to eliminate the constants via differentiation or other mathematical techniques.

9. Example 3 Eliminate the constant a from (x-a)2 + y2 = a2. (9)
Solution:
Thinking of y as a function of x and differentiating, we find
2(x-a) + 2 y dy/dx = 0.
Solving for a gives
a = x + y dy/dx. (10)
Solution (continued):
Substituting (10) into (9) for a gives:
Equation (11) can be simplified to the differential equation:
y2 = x2 + 2xy dy/dx (12)

10. Notes Another way to get (12) is to solve (9) for 2a: (x2 + y2)/x = 2a
and differentiate both sides. (Try this!)
It is much “easier” to start with a relation with constants and end up with a differential equation than it is to start with a differential equation and end up with a relation (or family of solutions)!

11. 1.4: Geometric Interpretation
To study the behavior of solutions of a differential equation, it is often useful to graph several members of a given family of solutions.
Example 4: Sketch some solutions of (5),
Note that through each point in the plane there passes one and only one member of the family of solutions! (Proof later …)

12. Example 5 Repeat with some solutions of (6), y’’ + y =0. Solution:
Try y1 = cos x + sin x (c1 = c2 = 1)
y2 = sin x (c1 = 0, c2 = 1)
y3 = -4 sin x (c1 = 0, c2 = -4)
Plotting these functions, we find that solutions cross many times!
For example, y2 and y3 cross at multiples of .
Therefore, the property of solutions we saw in Example 4 doesn’t hold here.
It turns out a different uniqueness property holds here – through each point in the plane passes one and only one member of the family of solutions that has a given slope! (Proof later …)

13. Example 6
Sketch a few members of the one-parameter family of curves given by (9), (x-a)2 + y2 = a2.
Solution:
This family gives implicit solutions to differential equation (12):
For example, the solution through (2,2) is
valid for 0<x<4.

14. Example 6 (continued)
Note that at any point (x,y) such that xy  0, (12) gives the slope of the line tangent to the curve through (x,y).
Also, any point (x,y) where xy  0 has a unique member of the family passing through it!
Question: What’s “wrong” with xy = 0?
Answer: Rewriting the differential equation (12) gives:
Therefore, infinite slope at points where xy =0.