Differential Equations
and Sequences/Series
with correct answers
2025/2026
Seperation of Variables for Differential Equations - correct answersKnow k is a relative growth
rate, y'-ky, more generally k*P, then you will get y=Ce^kt, Change in population is proportional
to the population, law of uninhibited growth
Change in quanittiy proportional to population
Basically solve it out with differential equation, substitute seperate integrate
Harder differential equations too, treat as differentials, then after doing that you have an inital
value with which you can use to find the overall c value for the equation, combine the c's
For the differential equations, will have many different solutions, and so when integrating may
lose a solution, see if the derivative is zero with that constant function and then if that constant
function can fit as the equation takes into account all numbers except for a specific sontant
Differentiability implies continuity
Must be able to put in the form f(y)dy=g(x)dx
Remeber there may be absolute values here too, specifically with the natural logs,
, Slope Fields - correct answersDo them sytematically, drawing a ton of little tangent lines around
specific points, then can connect to create an equation
equilibirum solutions are the constant function, easy to spot
Euler's Method - correct answersBasiallly approximate values to solutions of inital value
problems, derivative and some initial value, then use tangent lines ot move incrementally, more
steps more accurate, take existing x value and add on some certain amount, find the derivative
and you know the dx, so you can find the dy, with the dy then you add onto the y value, USE
THE TABLE
Kinda like a staircase
Depending on if increasing or decreasing then the tangent lines can be an underestimate or
overestimate
Logistic Growth Model - correct answersBasically see that there is some carrying capacity or
limit, like a rumor spreading, can't infintely spread
dP/dt = kP(1-P/L), get it into this form, if put in the carrying capacity, the derivative is then 0,
Kinda proportional, but as the population approaches L, then you have less and less growth,
how much room is there left
kP/L (L-P)
Derivative is like upside down quadratic
Zeros are at 0 and L, no growth at all, this is for the population, relative growth rate is (dP/dt)/P
= k-kP/L
Not constant as it is relative
Most rapid growth at the maximum of L/2 This is an inflection point
Recognize Logistic Growth model too
Zeros of this equation are the carrying capacity and 0
When doing the integral, use partial fractions to solve it out
get
P = L/(1+Ae^(-kt)), A = -1+L/Po
This is the Solution to the logsitic differential equation
and Sequences/Series
with correct answers
2025/2026
Seperation of Variables for Differential Equations - correct answersKnow k is a relative growth
rate, y'-ky, more generally k*P, then you will get y=Ce^kt, Change in population is proportional
to the population, law of uninhibited growth
Change in quanittiy proportional to population
Basically solve it out with differential equation, substitute seperate integrate
Harder differential equations too, treat as differentials, then after doing that you have an inital
value with which you can use to find the overall c value for the equation, combine the c's
For the differential equations, will have many different solutions, and so when integrating may
lose a solution, see if the derivative is zero with that constant function and then if that constant
function can fit as the equation takes into account all numbers except for a specific sontant
Differentiability implies continuity
Must be able to put in the form f(y)dy=g(x)dx
Remeber there may be absolute values here too, specifically with the natural logs,
, Slope Fields - correct answersDo them sytematically, drawing a ton of little tangent lines around
specific points, then can connect to create an equation
equilibirum solutions are the constant function, easy to spot
Euler's Method - correct answersBasiallly approximate values to solutions of inital value
problems, derivative and some initial value, then use tangent lines ot move incrementally, more
steps more accurate, take existing x value and add on some certain amount, find the derivative
and you know the dx, so you can find the dy, with the dy then you add onto the y value, USE
THE TABLE
Kinda like a staircase
Depending on if increasing or decreasing then the tangent lines can be an underestimate or
overestimate
Logistic Growth Model - correct answersBasically see that there is some carrying capacity or
limit, like a rumor spreading, can't infintely spread
dP/dt = kP(1-P/L), get it into this form, if put in the carrying capacity, the derivative is then 0,
Kinda proportional, but as the population approaches L, then you have less and less growth,
how much room is there left
kP/L (L-P)
Derivative is like upside down quadratic
Zeros are at 0 and L, no growth at all, this is for the population, relative growth rate is (dP/dt)/P
= k-kP/L
Not constant as it is relative
Most rapid growth at the maximum of L/2 This is an inflection point
Recognize Logistic Growth model too
Zeros of this equation are the carrying capacity and 0
When doing the integral, use partial fractions to solve it out
get
P = L/(1+Ae^(-kt)), A = -1+L/Po
This is the Solution to the logsitic differential equation
