Solve the system of equations using substitution.

Use undetermined coefficient to determine the solution of:y"-3y'+2y=2x+ex+2xex+4e3x

Kitty [74]

First check the characteristic solution: the characteristic equation for this DE is

r ² - 3r + 2 = (r - 2) (r - 1) = 0

with roots r = 2 and r = 1, so the characteristic solution is

y (char.) = C₁ exp(2x) + C₂ exp(x)

For the ansatz particular solution, we might first try

y (part.) = (ax + b) + (cx + d) exp(x) + e exp(3x)

where ax + b corresponds to the 2x term on the right side, (cx + d) exp(x) corresponds to (1 + 2x) exp(x), and e exp(3x) corresponds to 4 exp(3x).

However, exp(x) is already accounted for in the characteristic solution, we multiply the second group by x :

y (part.) = (ax + b) + (cx ² + dx) exp(x) + e exp(3x)

Now take the derivatives of y (part.), substitute them into the DE, and solve for the coefficients.

y' (part.) = a + (2cx + d) exp(x) + (cx ² + dx) exp(x) + 3e exp(3x)

… = a + (cx ² + (2c + d)x + d) exp(x) + 3e exp(3x)

y'' (part.) = (2cx + 2c + d) exp(x) + (cx ² + (2c + d)x + d) exp(x) + 9e exp(3x)

… = (cx ² + (4c + d)x + 2c + 2d) exp(x) + 9e exp(3x)

Substituting every relevant expression and simplifying reduces the equation to

(cx ² + (4c + d)x + 2c + 2d) exp(x) + 9e exp(3x)

… - 3 [a + (cx ² + (2c + d)x + d) exp(x) + 3e exp(3x)]

… +2 [(ax + b) + (cx ² + dx) exp(x) + e exp(3x)]

= 2x + (1 + 2x) exp(x) + 4 exp(3x)

… … …

2ax - 3a + 2b + (-2cx + 2c - d) exp(x) + 2e exp(3x)

= 2x + (1 + 2x) exp(x) + 4 exp(3x)

Then, equating coefficients of corresponding terms on both sides, we have the system of equations,

x : 2a = 2

1 : -3a + 2b = 0

exp(x) : 2c - d = 1

x exp(x) : -2c = 2

exp(3x) : 2e = 4

Solving the system gives

a = 1, b = 3/2, c = -1, d = -3, e = 2

Then the general solution to the DE is

y(x) = C₁ exp(2x) + C₂ exp(x) + x + 3/2 - (x ² + 3x) exp(x) + 2 exp(3x)

4 0

2 years ago

To legally drive a off-road vehicle, your age, a, must be 13 or over. Which of the following inequalities represents this senten

seropon [69]

Answer:

1.

Step-by-step explanation:

The > symbol shows that you have to be older than 13, and there must be a line underneath to signify a can be equal to the number (meaning you can be 13 to do this).

7 0

3 years ago

Read 2 more answers

What's the surface area ratio & the volume ratio?? please help me ASAP!!

Ulleksa [173]

Answer:

Step-by-step explanation:

Volumes of two spheres A and B = 648 cm³ and 1029 cm³

Things to remember:

1). Scale factor of two objects = $\frac{r_1}{r_2}$ [ $r_1$ and $r_2$ are the radii of two circles]

2). Area scale factor = $\frac{(r_1)^2}{(r_2)^2}$

3). Volume scale factor = $\frac{(r_1)^3}{(r_2)^3}$

Volume scale factor Or Volume ratio = $\frac{V_A}{V_B}$

$\frac{(r_1)^3}{(r_2)^3}= \frac{648}{1029}$

$\frac{r_1}{r_2}=\sqrt[3]{\frac{648}{1029} }$

$\frac{r_1}{r_2}=\frac{6(\sqrt[3]{3})}{7(\sqrt[3]{3})}$

$\frac{r_1}{r_2}=\frac{6}{7}$

Therefore, scale factor = $\frac{r_1}{r_2}=\frac{6}{7}$

≈ 6 : 7

Area scale factor Or area ratio = $(\frac{r_1}{r_2})^2=(\frac{6}{7})^2$

= $\frac{36}{49}$

≈ 36 : 49

Volume scale factor or Volume ratio = $\frac{648}{1029}$

= $\frac{216}{343}$

≈ 216 : 343

4 0

3 years ago

You finish washing 3/4 of your windows in an hour and a half. At that rate, how much longer will it take you to finish washing a

Marysya12 [62]

It takes you 90 minutes to wash 3/4 of your windows

Multiply both numbers by 4.

It takes you 360 minutes to wash 3 of your windows.

Divide both numbers by 3.

It takes you 120 minutes to wash 1 of your windows.

120 minutes = 2 hours.

It will take you 2 hours to wash all the windows.

5 0

3 years ago

Read 2 more answers

Need help please :) I’m not sure how to solve this one either

Andrews [41]

The answer is

580|21

6 0

1 year ago

Read 2 more answers