Question

Numbers 3 thru 4 please

Assignment 12. Introduction to Nonhomogeneous Equations
Read 4.4 Hand in the following problems:
If L is a linear operator, an equation of the form Ly = 0 is called a homogeneous equation and an equation of the form Ly = f is called a non homogeneous equation.
The solution of Ly = 0 is related to the solution of Ly = f. When you solve each of the following equations, look for the relationship between these solutions.

Rend 4.4 Hand in the following problems: If L is a linear operator, an oquation of the form Ly 0 is called a homogencous equation and an equstion of the form Ly-f is called a nonhomogeneous eguation. The solution of Ly = 0 is related to the solution of Ly f, when you solve each of the following equations, look for the relationship between these solutions la. (D-2), O 1b. 2a dr dy da 2b 3. You can roduce the following socond order equations to first order equations by making the substitutiony. Solve ench of the following. 3a. 4. Define A, x, 0 and b as follows -2 2 0 -4 Use matrix reduction (or any legitimate method) to solve each of the following matrix 4a Ax -0 1b.

Answer 1

Y^ - y^ = 0 Rightarrow u' - u = 0 {put y' = u - (1) y^ = u' Rightarrow du/dx = u Rightarrow du/u = dx integrating both sides Rightarrow log u = x + c Rightarrow u = e^x + x = Ae^x Rightarrow y' = Ae^x (from (1)) Rightarrow integral dy = A integral e^x dx Rightarrow y = Ae^x + B y^ - y' = e^x Rightarrowu' - u = e^x {put y' = u - (1) y^ = u' Rightarrow du/dx - u = e^x If: e^-integral 1 dx = e^-x Rightarrow u e^-x = integral e^x e^-x dx + c = integral 1 dx + c = x + c Rightarrow u = (x + c) e^x Rightarrow y' = (x + c) e^x (from (1)) Rightarrow integral dy = integral (xe^x + ce^x) dx Rightarrow y = x e^x - e^x + ce^x + A Rightarrow y = e^x (x + c - 1) + A