Setting

in the Euler equation we obtain

the general solution of the given equation is

The solution is quite straightforward. The given equation is a separable differential equation so

The correct answer is E.

By the formula we have

Let us calculate

and from integration by parts we have

Therefore, we obtain

The right answer is C.

Employing the properties of the integral,

Using the information supplied in the question, we obtain

We know that if the original mass was M₀, then after 15 days we have 

Hence

Thus we obtain

The general solution of the ODE is

Substituting

we obtain

This equation is quadratic in y and we obtain

Only

satisfy the given initial condition.

The Gini index is calculated using the formula

Where

is the Lorenz curve, Inserting

given in

Correct answer is  D .

Since

and 

The area under the graph of the supply function corresponds to the expected income. So we have

The right answer is E.

Let us use the change of variables method. Let

which gives

and for upper bound

and for lower bound

Substituting these values into the integral we get

The correct answer is A.

Let u=x³+x that gives du=(3x²+1)dx and 

we substitute and in the integral we have

We first find the intersection points of the given curves. Let us equate them

We find the intersection points

For the interval

we have 

so the area A is evaluated as

The correct answer is C.

To use the integration by parts method, let u=ln3x and dv=6x²dx.  Then we have 

and v=2x³ Inserting above equalities into ∫udv=uv-∫vdu we get;

where c is an arbitrary real number.

the correct answer is D.

Let

Therefore the correct answer is the C option.

Correct answer is  A.

Since the function in question is a linear one, the integral may be calculated by using the areas of triangles and squares. For –3≤x≤0, let A1 denote the area of the triangle above the x-axis and under the graph. In the interval [0, 2], the area is denoted by A2, and for 2≤x≤4, the area under the graph is denoted by A3. Then the definite integral is found as 

Therefore the correct answer is the D option.

The integrating factor is

Multiplying the equation

by this factor leads to the equation

By integrating both sides of above equation, using integration by parts on the second term on the right side, we obtain

where c is an arbitrary constant. Thus