Solve the given differential equation by an appropriate substitution.

y(x) = -x sqrt(c_1 x - 1) or y(x) = x sqrt(c_1 x - 1)

Step-by-step explanation:

Solve Bernoulli's equation ( dy(x))/( dx) = (x^2 + 3 y(x)^2)/(2 x y(x)):

Rewrite the equation:

( dy(x))/( dx) - (3 y(x))/(2 x) = x/(2 y(x))

Multiply both sides by 2 y(x):

2 ( dy(x))/( dx) y(x) - (3 y(x)^2)/x = x

Let v(x) = y(x)^2, which gives ( dv(x))/( dx) = 2 y(x) ( dy(x))/( dx):

( dv(x))/( dx) - (3 v(x))/x = x

Let μ(x) = e^( integral-3/x dx) = 1/x^3.

Multiply both sides by μ(x):

(( dv(x))/( dx))/x^3 - (3 v(x))/x^4 = 1/x^2

Substitute -3/x^4 = d/( dx)(1/x^3):

(( dv(x))/( dx))/x^3 + d/( dx)(1/x^3) v(x) = 1/x^2

Apply the reverse product rule f ( dg)/( dx) + g ( df)/( dx) = d/( dx)(f g) to the left-hand side:

d/( dx)(v(x)/x^3) = 1/x^2

Integrate both sides with respect to x:

integral d/( dx)(v(x)/x^3) dx = integral1/x^2 dx

Evaluate the integrals:

v(x)/x^3 = -1/x + c_1, where c_1 is an arbitrary constant.

Divide both sides by μ(x) = 1/x^3:

v(x) = x^2 (c_1 x - 1)

Solve for y(x) in v(x) = y(x)^2:

y(x) = -x sqrt(c_1 x - 1) or y(x) = x sqrt(c_1 x - 1)

answer: a. 13/44

step-by-step explanation: judging by the information given to us, we are looking for experimental probability. we can see that 13 males have chosen chocolate ice cream. there are 44 ice creams in total (remember that the total is always the denominator). so our answer is 13/44.

delaney would like to make a 5lb nut mixture that is 60% peanuts and 40% almonds.

she has several pounds of peanuts and several pounds of a mixture that is 20% peanuts and 80% almonds.

let 'p' represent the number of pounds of peanuts needed to make the new mixture, and  

let 'm' represent the number of pounds of the 80% almond-20% peanut mixture.

:

p + m = 5

therefore

m = (5-p)

write an equation based on the percent of peanuts

1p + .2m = .6(5)

replace m with (5-p)

p + .2(5-p) = .6(5)

p + 1 - .2p = 3

p - .2p = 3 - 1

.8p = 2

p = 2/.8

p = 2.5 lbs of peanuts required

then

5 - 2.5 = 2.5 lbs of the peanut/almond mixture required

:

a) what is the system that models this situation?

p + m = 5

1p + .2m = .6(5)