KFS

So dx/dt=2.67×10^(-7)y-10.67x, where y=(1-exp(-2.67×10^(-7)t)(2.51×10^6), is that correct? Or dx/dt=2.67×10^(-7)y-1,067×10^(-6)x? Yes I have done integrating factors. I have no problem solving these equations only setting up the problem.

Thanks for answer but how do I represent the amount of pollutant in the second lake as a differential equation? I understood the process in Example 4 but not with the information of both lakes in problem 15.

The problem I'm stuck is 15. I added Example 4 so you can see what problem 15 is all about. What I've been trying is: the amount of pollutant in the first lake is y=(1-e^(-2.67*10^(-7)t)(2.51*10^6). So I multiplied y by the rate entering the second lake which is the one leaving the first lake. Then I multiply 10.67 by the amount of pollutant in the second lake which is what I have to find through a linear first-order differential equation. Then I have dc/dt=10.67y+10.67c. Something's wrong but I don't know what. I do not need help solving this kind of differential equations where I need help is FINDING THE DIFFERENTIAL EQUATION TO SOLVE.

The problem says: The equation R(dQ/dt)+Q/C=E describes the charge Q on a capacitor, where R, C, and E are constants. (a) Find Q as a function of time if Q=0 at t=0. (b) How long does it take for Q to attain 99% of its limiting charge? I solved (a) and I got Q=-exp(-t/RC+ln(EC))+EC. But I don't know how to solve (b), I don't know what to equal Q to solve for t.

Okay I see it now. Thank you, I was stuck in this.

How did you get xy=sinh(x+y)?

The problem asks to find dy/dx in y=sinh(x+y)/xy=1. What I do is: differentiate both sides using implicit differentiation which gives d/dx(sinh(x+y)/xy)=0. I differentiate this setting y' where I have to differentiate y with respect to x. Thus I get (xycosh(x+y)+xyy'cosh(x+y)-ysinh(x+y)-xy'sinh(x+y))/(xy)^2. Then I solve for y' and I get y'=(-xycosh(x+y)+ysinh(x+y))/(xycosh(x+y)-xsinh(x+y)). But the answer in the book says y'=(y-cosh(x+y))/(cosh(x+y)-x). What am I doing wrong? Is my procedure incorrect? Thank you.

Is this correct?

How did you get the 500,000?

The formula is f'=-(kappa)×f, where f=f(0)e^(-kappa×t), or (1/kappa)×ln2. In this case what would each term mean? I didn't understand that.

How do I do it then? Can you explain?

The problem says: It takes 300,000 years for a certain radioactive substance to decay to 30% of its original amount. What is its half-life? The result is 173,000 years, but I don't see how it is obtained. I tried solving for x in f(x)=300,000e^(300,000×x)=0.3, which is approximately x=0.00004. Then I solved for x in f(x)=e^(-0.00004×x)=1/2, which is approximately x=173286. That's the closest I got to the actual result. What am I doing wrong? How do I obtain 173,000? Or maybe my answer is correct and 173,000 it's just an approximation? Thanks.

Thanks for your help. I just wanted to stick to the book procedure but sometimes the book makes it harder than it really is, this has happened before.

In Example 3 and Example 4 what is the meaning of omega? Why is it 1 and 3 respectively? I don't know how these results are obtained. I differentiated the expression above Example 3 and filled in with the given information in Example 3 but it didn't make anything clear. Thanks.