Solve Differential eq with initial conditions

[billiards]

Homework Statement

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.

 

[tiny-tim]

hi billiards!

I tried using Taylor series expansion …

eugh!

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

 

[billiards]

separate the variables

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?

 

[tiny-tim]

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

he he

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

nothing … they are the same to first order …

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