# Solve Differential eq with initial conditions

1. Oct 15, 2010

### billiards

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

Solve: $$\frac{\partial H}{\partial t} = -4\kappa H^{2}$$

With initial condition: $$H(0) = 1/L^{2}$$

To find: $$H(t) = \frac{1}{4\kappa t + L^{2}}$$

2. The attempt at a solution

I tried using Taylor series expansion such that:

$$H(t)\approx H(0)+t\frac{\partial H}{\partial t}(0)+.....$$

To first order this yielded: $$H(t)=\frac{L^{2}-4\kappa t}{L^{4}}$$

This is wrong unless t and/or k equals zero. Therefore this is wrong, it is not the general solution. Please help if you can. Thanks.

2. Oct 15, 2010

### tiny-tim

hi billiards!
eugh! :yuck:

just separate the variables: dH/H2 = -4k dt

3. Oct 15, 2010

### billiards

Thanks tiny-tim, with that hint I solved it straight away. You have no idea how long I was stuck.

Out of interest, what was wrong with my Taylor series approach?

4. Oct 15, 2010

### tiny-tim

he he
nothing … they are the same to first order …

what is the inverse of 1 + (4k/L2)t, to first order ?