Consider the differential equation (x + y + 1) dx + (2x + 2y + 1)dy = 0. Which of the following statement is true?a)A suitable substitution transform the differentiable equation to the variables separable fromb)The differential equation is linearc)The differential equation is exactd)ex + y is an integrating factor of the differential equationCorrect answer is option 'A'. Can you explain this answer?

Question

Answer

To determine which statement is true for the given differential equation, let's examine each option:

a) A suitable substitution transforms the differential equation to variables separable form.

To check if a suitable substitution can transform the differential equation to variables separable form, let's rewrite the given equation:

(x y 1) dx (2x 2y 1) dy = 0

Expanding this gives us:

x dx + y dx + dx = 2x dy + 2y dy + dy

Rearranging the terms gives:

(x dx - 2x dy) + (y dx - 2y dy) + dx - dy = 0

Factoring out the common factors:

x(dx - 2dy) + y(dx - 2dy) + dx - dy = 0

Now, notice that (dx - 2dy) is a common factor in the first two terms. Let's factor it out:

(dx - 2dy)(x + y) + dx - dy = 0

We can see that (dx - 2dy) is also a common factor in the last two terms. Let's factor it out:

(dx - 2dy)((x + y) + 1) = 0

Now we have a separable equation with the factors (dx - 2dy) and ((x + y) + 1). Therefore, a suitable substitution can be made to transform the differential equation to variables separable form. Hence, option 'a' is correct.

b) The differential equation is linear.

A linear differential equation is of the form dy/dx + P(x)y = Q(x). The given differential equation does not have this form, so option 'b' is not correct.

c) The differential equation is exact.

An exact differential equation is of the form M(x, y) dx + N(x, y) dy = 0, where ∂M/∂y = ∂N/∂x. Let's check if this condition is satisfied for the given differential equation:

∂M/∂y = 1
∂N/∂x = 2

Since ∂M/∂y is not equal to ∂N/∂x, the differential equation is not exact. Hence, option 'c' is not correct.

d) ex y is an integrating factor of the differential equation.

To check if ex y is an integrating factor, we need to multiply the given differential equation by ex y and see if it becomes exact. Let's do the multiplication:

(ex y)(x dx + y dx + dx) = (ex y)(2x dy + 2y dy + dy)

ex y(x dx + y dx + dx) = ex y(2x dy + 2y dy + dy)

ex y(x dx + y dx + dx) = 2ex y(x dy + y dy + dy)

ex y(x + y + 1) dx = 2ex y(x + y + 1) dy

Notice that the resulting equation is still not exact. Therefore, ex y is not an integrating factor. Hence, option 'd' is not correct.

In conclusion, the correct statement is option 'a': A suitable substitution transforms the differential equation to variables separable form.

Answer

Here (x + y + 1)dx + (2x + 2y + 1)dy = 0

By separables of variables,

Integrating (2z + log z) = x + c
⇒ 2(x + y) + log (x + y) = x + c.

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