General Contents
Detailed Contents
Index
Ordinary Differential Equations: Separation of Variables: Example 1
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Solve
by separation of variables.
Find both the general solution and the particular solution which passes through (1
,6
).
General Contents
Detailed Contents
Index
First, we find the general solution.
Rewrite the equation in differential form.
How can we “move” y from the right-hand side to the left-hand side?
We can divide both sides by y.
Do it.
We get
Are the variables separated?
No
.
We
need all of the x dependence on the right-hand side.
How can we accomplish this?
We can divide both sides by (x
3
+ 7).
Do it.
We get
Now the variables are separated.
What will the form of the integral on the left side be?
We will get
ln|y
|.
In order to integrate the right-hand side, we need to substitute u = (x
3
+ 7).
What will we get for du?
du = d(x
3
+ 7) = 3x
2
dx
.
How will the numerator on the right change?
x
2
dx
= du/3
Rewrite the differential equation in terms of u.
We get
Do the integration.
We get
Rewrite this result in terms of x, using the definition of u from above.
We get
We want to solve for y.
It will be easier if we replace C by
ln
(
C1), a different form of a constant.
Rewrite the equation using C1.
We can use some properties of logarithms here.
Change the 1/3 multiplier to an exponent.
Finally, replace the sum of logs on the right-hand side with the log of a product.
To complete solving for y we need to “undo” the
ln
function.
How do we do that?
We can use
Apply this here.
Find the general solution from this.
How do we find the particular solution which passes through (1
,6
)?
We substitute the coordinates into the general solution and solve for C1:
Do it.
State the particular solution.
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by separation of variables. Find both the general solution and the
particular solution which passes through (1,6).