# A seemingly difficult differential equation

1. Jan 29, 2006

### hanson

Anyone know how to solve it?
Either step by step or using MatLab/Mathematica/Maple is ok.
$$\frac{a+bsinwt-c\sqrt{H}}{k}=\frac{dH}{dt}$$

I need it in modelling an engineering problem, but I simply don't have any idea to solve it...

2. Jan 29, 2006

### benorin

the solution is approaching ulgy

Maple and I can only solve it if b=0.

It is easy for a=b=0, e.g. the homogeneous case, namely

$$k\frac{dH}{dt}+c\sqrt{H(t)}=0$$

has the solution

$$H(t)=(C-\frac{ct}{2k})^2$$.

If $$a\neq 0,$$ then the solution is approaching ulgy, see for yourself: Implicitly it is given by

$$c^2t+ka\ln(-a^2+c^2H(t))+2ck\sqrt{H(t)}-ka\ln(a+c\sqrt{H(t)})+ka\ln(c\sqrt{H(t)}-a)+C = 0$$

have a nice day

Last edited: Jan 29, 2006
3. Jan 29, 2006

### hanson

thanks
but b cannot be zero.
anyone have more?

4. Dec 17, 2009

### r duckworth

I think that a solution is as follows:

H = ( a/c + (H0^2-a/c)exp(-c/2k t) + b c sin(wt)/(c^2+4*k^2*w^2) - 2 k b w cos(wt)/(c^2+4*k^2*w^2) )^2

Last edited: Dec 17, 2009