Can this differential equation be solved using separation of variables? (dy)/(dx)=(sin(y)+y)/(4x+7) Choose 1 answer: (A) Yes (B) No

Examine Differential Equation: First, let's examine the differential equation to see if it can be rearranged into a form where all terms involving y (and dy) are on one side and all terms involving x (and dx) are on the other side. The differential equation is given by: (dy)/(dx) = (sin(y) + y) / (4x + 7)Use Separation of Variables: To use separation of variables, we need to be able to write the equation in the form of g(y)dy = f(x)dx, where g(y) is a function of y only and f(x) is a function of x only. Let's try to separate the variables by multiplying both sides by (4x + 7)dx and dividing by (sin(y) + y): (4x + 7) * (dy) = (sin(y) + y) * (dx)Isolate Variables: Now, we attempt to isolate dy and dx on opposite sides: dy / (sin(y) + y) = dx / (4x + 7)Successful Separation: We have successfully separated the variables with y's on one side and x's on the other. This means that the differential equation can be solved using separation of variables, as long as the integrals of 1 / (sin(y) + y) with respect to y and 1 / (4x + 7) with respect to x are solvable.

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Can this differential equation be solved using separation of variables?

(dy)/(dx)=(sin(y)+y)/(4x+7)
Choose 1 answer:
(A) Yes
(B) No

Can this differential equation be solved using separation of variables? $\newline$ $\frac{d y}{d x}=\frac{\sin (y)+y}{4 x+7}$ $\newline$ Choose $1$ answer: $\newline$ (A) Yes $\newline$ (B) No

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Q. Can this differential equation be solved using separation of variables?

\newline

\frac{d y}{d x}=\frac{\sin (y)+y}{4 x+7}

\newline

Choose

1

answer:

\newline

(A) Yes

\newline

(B) No

Examine Differential Equation: First, let's examine the differential equation to see if it can be rearranged into a form where all terms involving $y$ (and $dy$ ) are on one side and all terms involving $x$ (and $dx$ ) are on the other side. The differential equation is given by: $\newline$ $\frac{dy}{dx} = \frac{\sin(y) + y}{4x + 7}$
Use Separation of Variables: To use separation of variables, we need to be able to write the equation in the form of $g(y)\,dy = f(x)\,dx$ , where $g(y)$ is a function of $y$ only and $f(x)$ is a function of $x$ only. Let's try to separate the variables by multiplying both sides by $(4x + 7)\,dx$ and dividing by $(\sin(y) + y)$ : $\newline$ $(4x + 7) \cdot (dy) = (\sin(y) + y) \cdot (dx)$
Isolate Variables: Now, we attempt to isolate $dy$ and $dx$ on opposite sides: $\newline$ $\frac{dy}{\sin(y) + y} = \frac{dx}{4x + 7}$
Successful Separation: We have successfully separated the variables with $y$ 's on one side and $x$ 's on the other. This means that the differential equation can be solved using separation of variables, as long as the integrals of $\frac{1}{(\sin(y) + y)}$ with respect to $y$ and $\frac{1}{(4x + 7)}$ with respect to $x$ are solvable.