FP3 Flashcards
(25 cards)
What are the 2 important limits?
As x -> infinity, x^k.e^-x -> 0 for any real value k
As x -> 0+, ,x^k.ln(x) -> 0- for all k>0
What are the 4 equations that allow conversion from polar to cartesian coordinates and vice versa?
x^2 + y^2 = r^2 x = r cosO y = r sinO tanO = y/x - be careful, draw a sketch! O is theta
What is the polar equation of a circle, centre (a,0), radius a?
r = 2acosO
What is the polar equation of a circle, centre at pole, radius a?
r=a
What is the polar equation of a half line through pole, angle alpha from initial line?
O=alpha
What is the solution to the differential equation dy/dx = ky?
Ce^kx
How do you solve differential equations in the form:
dy/dx + Py = Q where P and Q are functions of x?
Integrating factor method. Multiply equation by function I, where I = e^(integral P dx). Can then express left hand side of equation as d/dx(Iy) (product rule), and integrate both sides to find y.
What must you remember when using the integrating factor method of solving a differential equation?
Equation in standard form before starting (dy/dx + Py =Q)
Multiply standard form by I, not original form
How do you solve differential equations in the form:
ady/dx + by = f(x) where a and b are constants?
Complementary function and Particular integral method:
Write reduced equation, ady/dx + by = 0 and find a general solution to this, y(c), the complementary function
Find a particular solution to the full equation, y(p), the particular integral by inserting the appropriate trial function and solving to find the constants
General solution to full equation is given by y = y(c) + y(p)
When using the complementary function and particular integral method for solving first order differential equations (where the coefficients of dy/dx and y are constants), what should you try as the PI when f(x) = Ce^kx?
Try y(p) = ae^kx, where a is a constant to be found. But, if complementary function has the same exponential form as f(x), try y(p) = axe^kx
When using the complementary function and particular integral method for solving first order differential equations (where the coefficients of dy/dx and y are constants), what should you try as the PI when f(x) = Ccoskx or Csinkx
Try y(p) = acoskx + bsinkx (find a and b)
When using the complementary function and particular integral method for solving first order differential equations (where the coefficients of dy/dx and y are constants), what should you try as the PI when f(x) is a polynomial of degree n?
Try y(p) = ax^n + bx^n-1 +… (find a, b…)
How do you solve a differential equation of the form:
ad^2y/dx^2 +bdy/dx +cy =0?
Take the auxiliary equation - quadratic equation with same coefficients:
ak^2 + bk + c=0 and find roots. From roots, find appropriate general solution.
If the auxiliary equation has 2 distinct real roots, k(1) and k(2), what is the general solution to the second order differential equation?
y = Ae^k(1)x + Be^k(2)x
If the auxiliary equation has 2 real roots, k(1) = k(2), what is the general solution to the second order differential equation?
y = (Ax + B)e^k(1)x
If the auxiliary equation has 2 non-real roots, k(1) = p + qi; k(2) = p - qi, what is the general solution to the second order differential equation?
y = e^px(Acosqx + Bsinqx)
What is the solution to the differential equation:
d^2y/dx^2 + n^2y = 0?
Auxiliary equation: k^2 + n^2 = 0 (roots +/- ni)
General solution: y = Acosnx + Bsinnx
How do you solve a differential equation in the form:
ad^2y/dx^2 + bdy/dx + cy = f(x)?
Complementary function and particular integral method
Find general solution of reduced equation, ad^2y/dx^2 + bdy/dx + cy = 0 using auxiliary equation, giving complementary function, y(c)
Find particular integral of full equation by inserting the appropriate trial function giving particular integral, y(p)
General solution of full equation is given by y = y(c) + y(p)
When using the complementary function and particular integral method to solve second order differential equations (where the coefficients of d^2y/dx^2, dy/dx and y are constants), what should you try as the PI when f(x) = Ce^qx?
First look at CF: does it have a term in (i) Be^qx or (ii) Bxe^qx? If no, try y(p) = ae^qx If type (i), try y(p) = axe^qx If type (ii), try y(p) = ax^2e^qx
When using the complementary function and particular integral method to solve second order differential equations (where the coefficients of d^2y/dx^2, dy/dx and y are constants), what should you try as the PI when f(x) = Ccosqx or Csinqx?
First look at CF: is it in form Acosqx + Bsinqx?
If no: try y(p) = acosqx + bsinqx
If yes, try y(p) = axcosqx if f(x) = Csinqx or try y(p) = axsinqx if f(x) = Ccosqx
When using the complementary function and particular integral method to solve second order differential equations (where the coefficients of d^2y/dx^2, dy/dx and y are constants), what should you try as the PI when f(x) is a polynomial of degree n?
First, look at differential equation: is there a y term?
If yes, try y(p) = ax^n + bx^(n-1) +….
If no, try y(p) = ax^(n+1) + bx^n + cx^(n-1) +…
How do you solve second order differential equations with variable coefficients, such as:
d^2y/dx^2 + Pdy/dx + Qy = R, where P, Q and R are functions of x?
Substitute something in to get it to a form we can work with - substitution will always be given
When is it appropriate to use the integrating factor method of solving differential equations?
When you have a first order differential equation with variable coefficients of y
When is it appropriate to use the complementary functions and particular integral method of solving differential equations?
First order differential equations where the coefficients of dy/dx and y are constants
Second order differential equations where the coefficients of d^2y/dx^2, dy/dx and y are constants and the differential equation equals a function of x (not 0)
