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Introduction to Differential Equations - At A Glance:

Way back in algebra we learned that a solution to an equation is a value of the variable that makes the equation true. This is backwards kind of thinking we need for differential equations.

To check if a number is a solution to an equation, we evaluate the left-hand side of the equation at that number, then evaluate the right-hand side of the equation at that number. If we get the same value for both sides of the equation, the number is a solution to the equation.

Example 1

Determine if x = 5 is a solution to the equation

x2 + 6 = 4x + 11.


Example 2

Determine if x = 4 is a solution to the equation

x2 + 6 = 4x + 11.


Example 3

Determine whether the function y = x2 is a solution to the d.e.

xy' = 2y.


Example 4

Determine whether the function y = x2 + 4 satisfies the d.e.

xy' = 2y.


Example 5

Show that the function y = xex is a solution to the d.e.


Example 6

Show that y = x ln x is not a solution to the d.e.


Exercise 1

Determine whether y = ex is a solution to the d.e.

y' + y" = 2y.

Exercise 2

Determine whether y = xex is a solution to the d.e.

y' = xy.

Exercise 3

Determine whether P = e-t is a solution to the d.e.

Exercise 4

Determine whether y = x2 is a solution to the d.e.

Exercise 5

Determine whether y = xn is a solution to the d.e.

x2y" + ny = n2y.

Exercise 6

Show that y = 5x2 + 2 is a solution to the d.e.

Exercise 7

Show that y = 5x2 is not a solution to the d.e.

Exercise 8

Show that f (x) = sin x is a solution to the d.e.

f (2)(x) + f (x) = 0.

Exercise 9

Show that f (x) = exsin x is a solution to the d.e.

2f '(x) - 2f (x) = f (x).

Exercise 10

Show that y = ln x is a solution to the d.e.

ey = x2y'.

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