1. dy/dt + ty=y ; y(1)=3
y′′−8y′+16y=24x+2
Instructions:
Good day! Equations are given below. Please determine which of the following equation is not a homogeneous equation. Thank you!
Choose the letter.
a. (y² - xy) dx + x²dy = 0
b. y' = 2xy e x/y / x2+y² sin(x/y)
Letter b is in fraction form.
c. 3x2ydx + (y+x³) dy = 0
d. 2x2y+3y³ (x³ + 2xy²) y' = 0
The population of a certain bacteria follows the logistic growth pattern. Initially, there are 10 gram of bacteria present in the culture. Two hours later, the culture weighs 25 g. The maximum weight of the culture is 100 g.
A. Write the corresponding logistic model for the bacterial growth.
B. What is the weight of the culture after 5 hours?
C. When will the culture's weight be 75 g?
HOMOGENEOUS EQUATION
Solve the given equation below. Show complete solution.
xy' = x + y
When do we identify an equation as homogeneous?
Reduce the equation zxx - ( 1+ y2 ) zyy = 0 to canonical form
1. The eigenfunction equation X00 = -λX is regular since s(x)ρ(x) = 1 , 0 for all x.
2. Bessel’s equation is regular for x 2 [1; 2].
3. Bessel’s equation is singular for x 2 [0; 1].
1. Solve the Sturm-Liouville equation X00 = -λX on [0; π] with data X(0) = 0 and X(π) = 0,
Ans: eigenfunction X(x) = sin(nx) eigenvalue λ = n2, for any positive integer n.
2. Solve the Sturm-Liouville equation X00 = -λX on [0; L] with data X(0) = 0 and X0(L)+hX(L) =
0.
( Ans: eigenfunction X(x) = sinx pλ eigenvalue λ given by a positive solution of the transcendental equation pλ + h tanL pλ = 0.
y''-x2y'-2xy=0;y(0)=1&y'(0)=0 solve by reduction of order
2) Solve the following system
𝑑𝑥
𝑑𝑡
= 𝑥 + 𝑦
𝑑𝑦
𝑑𝑡
= −2𝑥 − 𝑦