Im having trouble explaining why intergrals are well defined. For instance:
\int_{0}^{\infty} \frac{1}{(x + 16)^{\frac{5}{4}}}dx.
Here do i say something like:
\mbox{The integral behaves at zero, and at } \infty, (x + 16)^{\frac{5}{4}} > x^{\frac{5}{4}} \mbox{ therefore the integral diverges.}
This is annoying me as i have the answer on the tip of my pen, just can't write it down. I'm not 100% sure i understand what the question is asking me to do.
Consider the quantity u = e^{-xy} where (x,y) moves in time t along a path:
x = \cosh{t}, \mbox{ } y = \sinh{t}
Use a method...
Another example of this question type that I am stuck on is:
\ddot{x} + 7\dot{x} + 12x = 24
Find the particular solution where x(0) = 3, \left \dot{x}(0) = -2
My working is:
Homogeneous:
For this we read off the characteristic quadratic:
\lambda^2 + 7\lambda + 12 = 0
Find...
Ah rite i see how that works. If i get two solutions for lambda, is my complimentary function:
Ae^{\lambda_1} + Be^{\lambda_2}
or
Ae^{\lambda_1 + \lambda_2} ?
I found a similar question in a past paper that has solutions and from it learned things about the characteristic equation you mentioned, and also to use A + B.
This question is: \ddot{x} + 4\dot{x} + 5x = 34e^{2t}
You get the characteristic quadratic: \lambda^2 + 4\lambda + 5 = 0
So...
\ddot{x} + 10\dot{x} + 34x = 50e^{-t}
Im stuck on getting the homogeneous equation. I am used to equations with just first derivatives, not second derivatives as well. Usually, i'd set the left hand side equal to zero and say:
set \left x = Ae^{\lambda{t}} \left and \left so \left \dot{x} =...
I'd already worked this out but thanks for that explanation - I am sure it'l help in the future. I am revising for exams - it was just a silly hurdle in a question - so no, not homework. Thanks again:smile:
Im unsure on a very very basic differetiation i need for part of a question.
Quite simply - differentiate x/4. Thats it. Or x over any number - just never knew the rule. Is it simply 1? or 4? or 1/4?
I need to know for a question where iv to find the differential du for u = sin(x/y)...
Question: A spehrical planet of Radius R has a density p which depends on the distance r from its centre according to the formula
p = \frac{p_0}{1 + (r/R)^2}
where p_0 is a constant. By dividing the planet up into spherical shells of a small thickness dr, find the mass of the planet...
Here iv to show that integrals are well defined and find their values:
a) I = \left\int_{0}{4} \frac{x - 1}{\sqrt{16 - x^2}}dx
this one i can show its well defined ok (the domain will be positive as x will be between 0 and 4) but can't find the value. So far my work goes:
I =...
Ah sorry, of course! I was tired, wasn't thinking straight. Hand in time has passed now, but i was determined to get to the bottom of this. I can see now how to do it. Thanks very much!