Solve the differential equation
y'''+y''+y'+y=sinx+e^x+e^{-x}
In: Advanced Math
1) find the solution t the non-homogenous DE
y''-16y=3e5x , y(0)=1 , y'(0)=2
2)find the solution to the DE using cauchy-euler method
x2y''+7xy'+9y=0 , y(1)=2 , y'(1)=3
3)find the solution to the DE using Laplace
y''+8y'+16y=0 , y(0)=-1 , y'(0)=8
In: Advanced Math
1. For each permutationσ of {1,2,··· ,6} write the permutation matrix M(σ) and compute the determinant |m(σ)|, which equals sgn(σ).
(a) The permutation given by 1 → 2, 2 → 4, 3 → 3, 4 → 1, 5 → 6, 6 → 5.
(b) The permutation given by 1 → 5, 2 → 1, 3 → 2, 4 → 6, 5 → 3, 6 → 4.
In: Advanced Math
A.) State Product Rule, Quotient Rule, and Chain Rule.
B.) Prove Power Rule
C.) Prove Product rule:
-By definition of derivative
-By implicit differentiation
D.) Prove the Quotient Rule
-By definition of derivative
-By implicit differentiation
-By product rule and chain rule
In: Advanced Math
Please find f(x') and f(x") for all:
1) f(x) = 3(x2 -2x)3/4(7 - 5x2)6
2) f(x) = (y - x)4(y2 - x)3
3) f(x) = x7/3 + 16x3 + x
4) = f(x) = (x2 - x) / (y2 - y)
In: Advanced Math
Newton's Law of Cooling is based on the principle that the rate of change of temperature y'(t) of a body in an environment with ambient temperature A is proportional to the difference b/w the temperature y(t) of the body and the ambient temperature A, so that y'=k(A-y) for some constant k.
(1) Sketch the direction field for the equation y'=K(A-y) for k=1/10 and A=70degrees. Then sketch several solution curves based on starting values of y(0) both greater than and less than A. Discuss the implication of your graph in general, and as it relates to the long-term limiting temperature of a cup of hot coffee initially at 190degreeF and a cup of iced coffee initially at 40degreesF in a room with ambient temperature 70degreesF.
In: Advanced Math
Problem 7. Assume that a subset S of polynomials with real
coefficients has a property:
If polynomials a(x), b(x) are from S and n(x), m(x) are any two
polynomials with real coefficients, then polynomial a(x)n(x) +
m(x)n(x) is again in S. Prove that there is a polynomial d(x) from
S, such that any other polynomial from S is a multiple of d(x).
In: Advanced Math
a) Prove by induction that if a product of n polynomials is
divisible by an irreducible polynomial p(x) then at least one of
them is divisible by p(x). You can assume without a proof that this
fact is true for two polynomials.
b) Give an example of three polynomials a(x), b(x) and c(x), such
that c(x) divides a(x) ·b(x), but c(x) does not divide neither a(x)
nor b(x).
In: Advanced Math
In: Advanced Math
Show that, given any 3 integers, at least 2 of them must have the property that their difference is even.
In: Advanced Math
Post your answer to one (1) of the following questions:
Directly and completely answer the question(s).
Clearly and accurately explain your answer based on factual information.
Include examples, illustrations and/or applications in your answer(s).
In: Advanced Math
2.) By Theorem 3.23 of the text, the linear diophantine equation of the form ax + by = c has no integral solutions if c is not divisible by (a, b), the greatest common divisor of a and b. On the other hand if (a, b) divides c, then we can use the Extended Euclidean Algorithm to find integers s, t such that sa + tb = (a, b); multiplying through by the correct factor gives an integral solution x, y. Write a Mathematica procedure that solves any linear diophantine equation of the form ax + by = c, whenever it is solvable. You should invoke your Extended Euclidean Algorithm.
In: Advanced Math
x*y' '+y'-x*y=0
1> use Frobenious series solution to solve the ODE.
In: Advanced Math
Solve the linear programming problem by the method of corners.
Minimize | C = 2x + 5y | ||||
subject to | 4x | + | y | ≥ | 38 |
2x | + | y | ≥ | 30 | |
x | + | 3y | ≥ | 30 | |
x ≥ 0, y ≥ 0 |
In: Advanced Math
Use the method of Undetermined Coefficients to find the solution of the initial value value problem:
y'' + 8y' + 20y = 9cos(2t) - 18e-4t, y(0) = 5. y'(0) = 0
In: Advanced Math