5 Differential Equations With solutions
Example 1
For the differential equation, find (a) one solution and then (b) all solutions.
y ' = 4x3
Answer
This looks like it came from using the power rule on the function
y = x4.
This is a solution. Since the derivative of a constant is zero, adding a constant to the
function x4 doesn't change the derivative of that function. So we can write all solutions
by
y = x4 + C
where C is any constant.
Example 2
For the differential equation, find (a) one solution and then (b) all solutions.
y' = x5
Answer
When we take the derivative of a polynomial the degree drops by 1, so this almost
looks like it came from taking the derivative of the function y = x6. However, the
derivative of x6 is 6x5. To cancel out the extra 6 we get when taking the derivative, we
need to introduce a factor of . A solution to the differential equation could look like
this:
Example 1
For the differential equation, find (a) one solution and then (b) all solutions.
y ' = 4x3
Answer
This looks like it came from using the power rule on the function
y = x4.
This is a solution. Since the derivative of a constant is zero, adding a constant to the
function x4 doesn't change the derivative of that function. So we can write all solutions
by
y = x4 + C
where C is any constant.
Example 2
For the differential equation, find (a) one solution and then (b) all solutions.
y' = x5
Answer
When we take the derivative of a polynomial the degree drops by 1, so this almost
looks like it came from taking the derivative of the function y = x6. However, the
derivative of x6 is 6x5. To cancel out the extra 6 we get when taking the derivative, we
need to introduce a factor of . A solution to the differential equation could look like
this:
