Name and locate points in all four quadrants of the coordinate plane.

Use the method of undetermined coefficients to find the general solution to the de y′′−3y′ 2y=ex e2x e−x

djverab [1.8K]

I'll assume the ODE is

$y'' - 3y' + 2y = e^x + e^{2x} + e^{-x}$

Solve the homogeneous ODE,

$y'' - 3y' + 2y = 0$

The characteristic equation

$r^2 - 3r + 2 = (r - 1) (r - 2) = 0$

has roots at $r=1$ and $r=2$ . Then the characteristic solution is

$y = C_1 e^x + C_2 e^{2x}$

For nonhomogeneous ODE (1),

$y'' - 3y' + 2y = e^x$

consider the ansatz particular solution

$y = axe^x \implies y' = a(x+1) e^x \implies y'' = a(x+2) e^x$

Substituting this into (1) gives

$a(x+2) e^x - 3 a (x+1) e^x + 2ax e^x = e^x \implies a = -1$

For the nonhomogeneous ODE (2),

$y'' - 3y' + 2y = e^{2x}$

take the ansatz

$y = bxe^{2x} \implies y' = b(2x+1) e^{2x} \implies y'' = b(4x+4) e^{2x}$

Substitute (2) into the ODE to get

$b(4x+4) e^{2x} - 3b(2x+1)e^{2x} + 2bxe^{2x} = e^{2x} \implies b=1$

Lastly, for the nonhomogeneous ODE (3)

$y'' - 3y' + 2y = e^{-x}$

take the ansatz

$y = ce^{-x} \implies y' = -ce^{-x} \implies y'' = ce^{-x}$

and solve for $c$ .

$ce^{-x} + 3ce^{-x} + 2ce^{-x} = e^{-x} \implies c = \dfrac16$

Then the general solution to the ODE is

$\boxed{y = C_1 e^x + C_2 e^{2x} - xe^x + xe^{2x} + \dfrac16 e^{-x}}$

6 0

11 months ago

A jacket that originally sold for $60.00 was on sale for 10% off. When it didn't sell after several weeks, the sale price was di

sammy [17]

The final price of the jacket was $32.40 after both the 10% and 40% discount. The total discount is NOT equal to a 50% discount. When 10% off, the price was $54, bringing the next 40% off to a whole different number, not necessarily off the original price.

I don't know if that makes sense to you, sorry xD

7 0

2 years ago

What is the solution to this system of equations?

Eduardwww [97]

Answer:

x = -2, y = 3

Step-by-step explanation:

Given equations are,

4x + 5y = 7 ---------(1)

3x - 2y = -12 ----------(2)

Multiply equation (1) by 3

12x + 15y = 21 ------(3)

Multiply equation (2) by (-4)

-12x + 8y = 48 --------(4)

Now add equations (3) and (4),

(12x + 15y) + (-12x + 8y) = 21 + 48

(12x - 12x) + (15y + 8y) = 69

23y = 69

y = 3

From equation (2),

3x - 2(3) = -12

3x - 6 = -12

3x = 6 - 12

x = -2

Therefore, x = -2 and y = 3 will be the answer.

4 0

3 years ago

Read 2 more answers

Find limit as x approaches 1 for x-1/ x^2-3x+2

zysi [14]

I'm going to assume that you meant the following:

(x-1)

---------------

x^2-3x+2

Factoring the denominator, we get:

(x-1)

------------

(x-1)(x-2)

Notice that we have x-1 in both the numerator and the denominator. This factor cancels. HOWEVER: x can still not equal 1, which would lead to div. by zero. With the factor x-1 cancelled, we have 1 / (x-2). As x approaches 1 from either side, y approaches 1 / (1-2), or 1 / (-1), or -1. Thus, the limit exists and is -1.

8 0

2 years ago

20c -18c=6 <br> Solve for c <br> C=

Leni [432]

Answer:

c=3

Step-by-step explanation:

5 0

2 years ago

Read 2 more answers