Can this differential equation be solved using separation of variables? (dy)/(dx)=10-2y+9x Choose 1 answer: (A) Yes (B) No

Check Equation Form: To determine if the differential equation (dy)/(dx) = 10 - 2y + 9x can be solved using separation of variables, we need to see if we can express the equation in the form of (dy)/(g(y)) = f(x)(dx), where g(y) is a function of y only and f(x) is a function of x only.Rearrange Equation: We attempt to rearrange the equation to isolate terms involving y on one side and terms involving x on the other side. The given equation is (dy)/(dx) = 10 - 2y + 9x.Isolate Terms: We try to move the term involving y to the left side of the equation and the term involving x to the right side of the equation. This would give us (dy)/(dx) + 2y = 10 + 9x.Check Functionality: Now, we check if the left side of the equation is only a function of y and the right side is only a function of x. The left side, (dy)/(dx) + 2y, is not solely a function of y because it contains the derivative term (dy)/(dx). The right side, 10 + 9x, is solely a function of x.Final Conclusion: Since the left side of the equation is not solely a function of y, we cannot separate the variables as required for the method of separation of variables. Therefore, the differential equation cannot be solved using separation of variables in its current form.

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Can this differential equation be solved using separation of variables?

(dy)/(dx)=10-2y+9x
Choose 1 answer:
(A) Yes
(B) No

Can this differential equation be solved using separation of variables? $\newline$ $\frac{d y}{d x}=10-2 y+9 x$ $\newline$ 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}=10-2 y+9 x

\newline

Choose

1

answer:

\newline

(A) Yes

\newline

(B) No

Check Equation Form: To determine if the differential equation $(\frac{dy}{dx} = 10 - 2y + 9x)$ can be solved using separation of variables, we need to see if we can express the equation in the form of $(\frac{dy}{g(y)} = f(x)\,dx)$ , where $g(y)$ is a function of $y$ only and $f(x)$ is a function of $x$ only.
Rearrange Equation: We attempt to rearrange the equation to isolate terms involving $y$ on one side and terms involving $x$ on the other side. The given equation is $\frac{dy}{dx} = 10 - 2y + 9x$ .
Isolate Terms: We try to move the term involving $y$ to the left side of the equation and the term involving $x$ to the right side of the equation. This would give us $\frac{dy}{dx} + 2y = 10 + 9x$ .
Check Functionality: Now, we check if the left side of the equation is only a function of $y$ and the right side is only a function of $x$ . The left side, $\frac{dy}{dx} + 2y$ , is not solely a function of $y$ because it contains the derivative term $\frac{dy}{dx}$ . The right side, $10 + 9x$ , is solely a function of $x$ .
Final Conclusion: Since the left side of the equation is not solely a function of $y$ , we cannot separate the variables as required for the method of separation of variables. Therefore, the differential equation cannot be solved using separation of variables in its current form.