# How to find my ode45 equation in specific h

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esat gulhan on 20 Aug 2020
Edited: Alan Stevens on 21 Aug 2020
syms D g H Do
tspan = [0 120];
mgiren=0
Do=3;
D=2/10;
h0=h;
g=9.81;
y0 = 2;
ySol= ode45(@(t,h)(mgiren-(pi()*D^2/4*(2*g*h)^0.5)/(pi()*Do^2/4)), tspan, y0)
for t=linspace(0,100,11)
fprintf('%15.5f',t,deval(ySol,t)),;fprintf('\n')
end
My differantial code is here, dt/dh=(mgiren-(pi()*D^2/4*(2*g*h)^0.5)/(pi()*Do^2/4)),h(0)=2, h(tx)=1, how can i find tx, is there anyway to find tx, i can find it with interpolate but is there any easier way.

Alan Stevens on 21 Aug 2020
Edited: Alan Stevens on 21 Aug 2020
Something like this perhaps:
tspan = 0:120;
h0=2;
[t, h] = ode45(@rate, tspan, h0);
tx = interp1(h,t,1);
plot(t,h,tx,1,'o'), grid
text(tx+5,1,['tx = ' num2str(tx)])
xlabel('t'),ylabel('h')
function dhdt = rate(~, h)
mgiren=0;
Do=3;
D=2/10;
g=9.81;
dhdt = (mgiren-(pi*D^2/4*(2*g*h)^0.5)/(pi*Do^2/4));
end Ok, I guess this still uses interpolation!
You could use fzero to find it, but, for this curve I think interpolation is far simpler.

esat gulhan on 21 Aug 2020
Actually two ways are good for this. but there is a problem when i try to reach 0. interp(h,t,0) gives error, same the fzero gives error too.
Alan Stevens on 21 Aug 2020
Tht's because you can't get to zero with the data specified. There is an analytical solution to your equation, which is most easily expressed with t as a function of h - see below. You'll notice there is a logarithmic term, which tries to calculate log(0) when both mgiren and h are zero.
tspan = 0:120;
h0=2;
mgiren=0;
Do=3;
D=2/10;
g=9.81;
a = mgiren/(pi*Do^2/4);
b = (D/Do)^2*sqrt(2*g);
T = @(h) -2*(b*sqrt(h) + a*log(a-b*sqrt(h)))/b^2 ...
+ 2*(b*sqrt(h0) + a*log(a-b*sqrt(h0)))/b^2;
h = h0:-0.01:0.01;
t = T(h);
plot(t,h), grid
xlabel('t'), ylabel('h') Alan Stevens on 21 Aug 2020
Hmm. Thinking further, the log term is zeroed all the way through because it's multiplied by a (which is zero)., so, basically, you just have a square root relationship (when a is zero).
You get NaN if you try T(0).
T(h = 0) can be found from 2*sqrt(h0)/b;