calculus: differential equation?

Question

How would you solve the differential equation
y' = k(M-y)??
I need to make a solution curve out of this and I am confused??
Any help is greatly appreciated!!

any suggestions on how I would sketch the solution curve for this?

Answer 1

First change y' into dy/dx.
So you have dy/dx=k(M-y)
Separate dy and dx to obtain dy*1/(M-y)=kdx
Integrate both sides: -ln(M-y)=kx+C
M-y=Ce^(-kx) (Don't worry about the negative with the C because multiplying it by negative one is keeping it a constant.)
Solve for y: y=M-Ce^(-kx)

Answer 2

the first hassle-free rule is then you definitely in hassle-free words have x's and y's, take a million off the potential and circumstances the fashion infront of the x by the potential as an get jointly y=mx^2 + c dy/dx = 2mx (the c disappears) in an attempt to respond to the first question y=3x^4- 2.5x^3+ 4x^2- 3x+ 6 dy/dx = (3*4)x^(4-a million) - (2.5*3)x^(3-a million) + (4*2)x^(2-a million) - 3 = 12x^3 - 7.5x^2 + 8x -3 the 2d one is an similar in hassle-free words you opt to understand the regulations of powers. i advice understanding those. If x^0 = a million If x^a million = x if x^-a million = a million/x if x^a million/2 = SQRT x If x^3/2 = SQRT x^3 in reality, if one has x^a/b Take the b root of x^a now shall we do question 2 s= 3/2 SQRT x^5 + 4/3x^3 s = 3/2 x^5/2 + 4/3x^3 ds/dx = (3/2 * 5/2)x^(5/2-a million) + (4/3*3)x^(3-a million) = 15/4x^3/2 + 4x^2 question 3 is composed of sin and cos. the mandatory rule right here's in case you differentiate sin you get cos, and in case you differentiate cos you get -sin if x (used particularly of theta as i do no longer understand the thanks to operate that in) has a mode in the front of that one merely multiply that style by any style in the front of sin/cos now all of us have we opt to finish question 3 y=2sin3x - 4cosx dy/dx = 6cos3x +4sinx question 4 e is a particular style, once you differentiate it it remains itself. Eg, then y=e^x dy/dx = e^x even as y= me^nx dy/dx = mne^nx for lnx , it really is spinoff is a million/x extending this, the spinoff of ln(mx^n) = (spinoff of mx^n) / (mx^n) (you also opt to study log regulations, they don't look needed for this question yet for the more effective complicated ones they are. i visit educate you both ideas for this so that you understand the position it comes from. it really is the in hassle-free words one you opt to understand for this question mln(x) = ln(x^m) ) so, making use of this training we may be able to finish question 4 no longer making use of the log regulation y = 4e^2x + 3ln(x) dy/dx = 8e^2x + 3/x (merely take the fashion in the front of the ln x and positioned it on good. hassle-free. ) making use of the log regulation y = 4e^2x +3lnx = 4e^2x + ln(x^3) dy/dx = 8e^2x + (3x^2/x^3) = 8e^2x = 3/x (the powers cancel) i quite desire this helped. i advice writing them down on paper with my determining so that you'll visualise it and then artwork by them and verify that i have not lengthy previous incorrect. My psychological maths isn't tremendous so i ought to have made multiplying blunders.

Answer 3

Wow...I'm in calculus and I have no idea. If there were numbers in it I could tell you though.

Answer 4

dy/dx= k(M-y) so
dy/(M-y)=kdx
lnI M-yI=- kx+C soI M-y I= C_1 *e^-kx so C_1 must be >0 and
M-y = C_1e^-kx and y =M -C_1e^-kx
or y=C-1 e^-kx+M