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)

For every 84 pens there are 112 pencils in the box.

Step-by-step explanation:

We are given the following information in the question.

We are given a ratio of pens to pencil is 84 to 112.

Ratio is a relationship between two numbers that helps us to know how first number is related to the second number.

This can be written as: 84 : 112

This can be read as: For every 84 pens there are 112 pencils in the box.

Simplifying the fraction we have:

This ration can be written as 21 : 28 which means for every 21 pens in a box there are 28 pencils in the box.