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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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Is (x, y) a solution to the nonlinear equation?

Full solution

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.

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