Consider the differential equation ( x + y + 1) dx + (2x + 2y + 1) dy = 0. Which of the following statements is true?a)The differential equation is linearb)The differential equation is exactc)ex + y is an integrating factor of the differential equationd)A suitable substitution transforms the differentiable equation to the variable separable formCorrect answer is option 'D'. Can you explain this answer?

Question

Answer

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

a) The differential equation is linear:
A linear differential equation is one in which the dependent variable and its derivatives appear only to the first power and are not multiplied together. In the given equation, both x and y appear to the first power, but they are multiplied together in the terms (xy)dx and (2xy)dy. Therefore, the given differential equation is not linear.

b) The differential equation is exact:
A differential equation is exact if it can be written in the form M(x, y)dx + N(x, y)dy = 0, where M and N are functions of x and y, and ∂M/∂y = ∂N/∂x. Let's check if this condition holds for the given equation:
M(x, y) = x + y
N(x, y) = 2x + 2y
∂M/∂y = 1
∂N/∂x = 2
Since ∂M/∂y is not equal to ∂N/∂x, the given differential equation is not exact.

c) ex^y is an integrating factor of the differential equation:
An integrating factor is a function that can be multiplied to a differential equation to make it exact. In this case, the integrating factor would need to be able to transform the given equation into an exact one. However, as we determined in the previous option, the given equation is not exact. Therefore, ex^y is not an integrating factor.

d) A suitable substitution transforms the differential equation to the variable separable form:
Variable separable form means that the equation can be written in the form f(x)dx = g(y)dy, where f(x) and g(y) are functions of x and y, respectively. To determine if a suitable substitution can transform the given equation into this form, let's make the substitution x = u and y = v, where u and v are functions of a new variable t (u = u(t) and v = v(t)). Substituting these into the given equation, we have:
(uv)du + (2uv)dv = 0
Factoring out uv, we get:
uv(du + 2dv) = 0
Since this equation can be separated into f(u)du = -2g(v)dv, where f(u) = uv and g(v) = -v, we can conclude that a suitable substitution transforms the differential equation to the variable separable form.

Therefore, the correct answer is option d) A suitable substitution transforms the differential equation to the variable separable form.

Answer

This equaton can be identified by that constant 1
it can be reduce to variable separable form
or u can also check other options
just like
here dM/dy is not equal to dN/dx hence it is not exact

Answer

Op tion d

Answer

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

a) The differential equation is linear:
A linear differential equation is one in which the dependent variable and its derivatives appear only to the first power and are not multiplied together. In the given equation, both x and y appear to the first power, but they are multiplied together in the terms (xy)dx and (2xy)dy. Therefore, the given differential equation is not linear.

b) The differential equation is exact:
A differential equation is exact if it can be written in the form M(x, y)dx + N(x, y)dy = 0, where M and N are functions of x and y, and ∂M/∂y = ∂N/∂x. Let's check if this condition holds for the given equation:
M(x, y) = x + y
N(x, y) = 2x + 2y
∂M/∂y = 1
∂N/∂x = 2
Since ∂M/∂y is not equal to ∂N/∂x, the given differential equation is not exact.

c) ex^y is an integrating factor of the differential equation:
An integrating factor is a function that can be multiplied to a differential equation to make it exact. In this case, the integrating factor would need to be able to transform the given equation into an exact one. However, as we determined in the previous option, the given equation is not exact. Therefore, ex^y is not an integrating factor.

d) A suitable substitution transforms the differential equation to the variable separable form:
Variable separable form means that the equation can be written in the form f(x)dx = g(y)dy, where f(x) and g(y) are functions of x and y, respectively. To determine if a suitable substitution can transform the given equation into this form, let's make the substitution x = u and y = v, where u and v are functions of a new variable t (u = u(t) and v = v(t)). Substituting these into the given equation, we have:
(uv)du + (2uv)dv = 0
Factoring out uv, we get:
uv(du + 2dv) = 0
Since this equation can be separated into f(u)du = -2g(v)dv, where f(u) = uv and g(v) = -v, we can conclude that a suitable substitution transforms the differential equation to the variable separable form.

Therefore, the correct answer is option d) A suitable substitution transforms the differential equation to the variable separable form.

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