# Differential equation using partial fractions

1. Apr 13, 2007

### Shelby

1. The problem statement, all variables and given/known data

I need to integrate this differential equation using partial fractions to obtain an equation for P in terms of t; P(t):

1/P dP/dt = b + aP

2. Relevant equations

3. The attempt at a solution

So far, this is what I have:
ln /P/ = bP + aP^2/2 +c

Where do I go from here, would I take e from each side and then what would I do? and am I doing this right?

Last edited by a moderator: Apr 13, 2007
2. Apr 13, 2007

### christianjb

Multiply both sides by P?

3. Apr 13, 2007

### Curious3141

No sense in doing that, since he still needs to bring the P-terms over to the left hand side.

Bring the P-terms from the RHS to the LHS, then separate variables :

$$\frac{1}{P(aP+b)}dP = dt$$

Separate the left hand side using partial fractions (write it as $$\frac{k_1}{P} + \frac{k_2}{aP+b}$$, then find the values of k1 and k2 quickly with the "Heaviside cover-up" shortcut, for example) and integrate both sides to solve.

4. Apr 13, 2007

### HallsofIvy

Staff Emeritus
No. You can't just integrate both sides with respect to P: there is not "dP" on the right side. You can rewrite the equation as
$$\frac{dP}{P(b+aP)}= dt$$
and integrate- the left side with respect to P, the right side with respect to t.

As Curious3141 said, you "partial fractions" to integrate the right side.

5. Apr 13, 2007

### Shelby

partial fractions

can you point me in the right direction, with all of these variables I keep getting confused?

6. Apr 13, 2007

### HallsofIvy

Staff Emeritus
What more "pointing" do you want! Use partial fractions to integrate
$$\frac{dP}{P(b+ aP}$$

Can you do the "partial fractions" decomposition? You want to find A and B so that
$$\frac{1}{P(b+aP)}= \frac{A}{P}+ \frac{B}{b+ aP}$$
Multiply both sides of that by P(b+aP) to get
$$1= A(b+aP)+ BP$$
and solve for A and B. (Hint: let P= 0 and P= -b/a)