differential equations

review- fundamental theorems of calculus

#1 If f is continuous on [a,b] and F is an antiderivative of f on then...integral from a to b of f(x) dx = F(x)| b,a= F(b) - F(a) C is NOT needed- cancels out -find antiderivative, then plug into EQ #2 for integrally defined function, F(x)=∫a,x f(t)dt, where a is any number -use first fundamental theorem, except x's will be used instead of numbers, if you take the derivative of this, you get the original function but replaced with x-can plug in values for x after -d/dx[∫a,x f(t)dt]=f(x) OR d/dx[∫a,g(x) f(t)dt=f(g(x))*g'(x)

review- integral rules ∫a,b f(x)dx= ∫a,a f(x)dx= ∫a,b k*f(x)dx= ∫a,b [f(x)+/-g(x)]dx= ∫a,b f(x)dx + ∫b,c f(x)dx= ∫a,b [f(x)+k]dx= ∫a+h,b+h f(x-h)dx=

- ∫b,a f(x)dx 0 k ∫a,b f(x)dx ∫a,b f(x)dx+/- ∫a,b g(x)dx ∫a,c f(x)dx ∫a,b f(x)dx + ∫a,b kdx ∫a,b f(x)dx can shift left/right, convert either way

drawing slope fields -through points?

-for each coordinate, plug into given equation to get slope there, then draw slope (can use ruler, use other dots to draw parallel line) -slope 0=horizontal line, undefined=vertical line -you can start to see graphs form when you connect dots, can see asymptotes too -if given y equation, solve for derivative first -make sure to go through point, follow slope lines next to it, watch out for asymptotes

separating differential EQs- general and particular solutions

1.separate variables into x and y (transfer only what is necessary to dy side) *rmb y'=dy/dx*, can use multiplication/division, cross multiplying, exponent rules 2. integrate both sides, using rules/u-sub *make sure to include +C on one side), things in parantheses are often u-sub 3. isolate y or y^2 (or when not possible, C=constant) -leave y^2 alone if general solution -convert between e and ln if necessary particular solution -plug in coordinate (or given info) to general solution equation, solve for C, rewrite EQ with C -for y^2, solve for y, rmb it is +/- square root, look at y value of given coordinate to see *for ln, since it has absolute value signs, check if number was positive or negative when plugging in to see if you need to multiply by a negative* both? can find general solution, then C, then solve for y -if one part of equation you can't integrate ex. sin(x^2), can use fundamental theorem of calculus, to decide between answer choices, look at coordinate, plug in x value and evaluate for y

direct and inverse variation -writing equation

direct y=kx inverse y=k/x "reciprocal" -"the rate of change with _____ with respect to _____"= ex. dy/dx, or think of for exampes miles per hour to write, can change variables, if second part is not given, it is usually time -is... proportional/inversely proportional (write k- use EQs) to... (write out EQ) -acceleration= second derivative

"the rate of change of y is proportional to y" -solution? -how to do problems

dy/dt=ky solution (when you do separating variables method): y=Ce^kt, when you differentiate you get ky C=initial value k=rate t=time (can be other variable) -write out formula, use values given with 0, plug in for y and t to find C, rewrite formula with C, plug in other values, move C over, rewrite as ln, solve for k, rewrite formula with everything, then plug in t, put into calculator to solve -if not given multiple values, use what is given to solve for k

review u-sub ∫f'(g(x))* g'(x) dx=?

f(g(x))+C u=g(x) make it the largest piece you have/can make derivative for, include +/-then find du=g'(x) derivative optional- make dx=du/g'(x) or just write opposite of whatever you multiply by (should be a constant) outside of integral rewrite as ∫f(u)du then find antiderivative and rewrite as F(u)+C plug back in- F(g(x))+C

general vs particular solutions for differential equations

general- any function y=f(x) which satisfies the equation ex. y'=2x would be y=x^2+C, where C is any real number (F(x)+C) particular- satisfies a known initial condition as well (solving for C)

determining/verifying if functions are solutions to differential equations -includes problems like slope of y'=xy? -if asked graph passes through a point, plug into original EQ to check

if necessary, solve for derivatives of given equation (multiply out if needed), plug in for y, leave any x's, check if it equals other side of EQ, if yes then it is a solution

differential equation -x&y?

relationship between one or more variables and the rate of change of one or more of the variables -equation that involves x, y, and derivatives of y ex. 2xy'-3y=0

review- antiderivatives ∫dx ∫x^p dx ∫e^x dx ∫sin x dx ∫sec^2x dx ∫sec x tan x dx ∫1/√(1-x^2)=du/√1-u^2 ∫1/x√(x^2-1)dx=du/u√u^2-1 ∫k dx ∫1/x dx do NOT integrate as a power ∫a^x dx ∫cos x dx ∫csc^2x dx ∫csc x cot x dx ∫1/(1+x^2)= du/u^2+1 ∫cot x dx ∫tan x dx ∫sec x dx ∫csc x dx

x+ C 1/(p+1) (x^p+1) +C e^x+C -cos x +C tan x + C sec x +C arcsin(x) + C= arcsin(u)+C arcsec|x|+C= arcsec|u|+C kx+C ln|x|+C 1/ln a * a^x + C sin x +C -cot x +C -csc x + C arc tan (x) + C= arctan(u)+C ln|sin x|+C -ln|cos x|+C ln|sec x +tan x|+ C -ln|csc x+ cot x|+ C

review- converting logs and exponents ln(1)? ln(0)? e^0? ex. ln(e^6) -ln and exponent rules

y=e^x, then x=ln y ln(1)=0 ln(0)=undefined e^0=1 ln(ab)=ln(a)+ln(b) ln(a/b)=ln(a)-ln(b) ln(a^b)=bln(a) x^ax^b=x^a+b x^a/x^b=x^a-b (x^a)^b=x^ab (xy)^a=x^ay^a etc.