y^(')=2x+3y-1 Is y=-(2)/(3)x+(1)/(9) a solution to the above equation? Choose 1 answer: (A) Yes (B) No

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y^(')=2x+3y-1 Is  y=-(2)/(3)x+(1)/(9) a solution to the above equation? Choose 1 answer: (A) Yes (B) No

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Given Differential Equation: Given the differential equation y'=2x+3y-1, we need to check if the function y=-(2)/(3)x+(1)/(9) is a solution. To do this, we will first find the derivative of y with respect to x, which we denote as y'.Find Derivative of y: The derivative of y with respect to x is found by differentiating each term of y=-(2)/(3)x+(1)/(9) with respect to x. The derivative of -(2)/(3)x with respect to x is -(2)/(3), and the derivative of a constant, (1)/(9), is 0. Therefore, y' = -(2)/(3).Substitute y and y': Now we will substitute y and y' into the given differential equation to see if it holds true. Substituting y' = -(2)/(3) and y = -(2)/(3)x + (1)/(9) into the differential equation y'=2x+3y-1 gives us: -(2)/(3) = 2x + 3(-2/3x + 1/9) - 1.Simplify Right Side: We simplify the right side of the equation: 2x + 3(-2/3x + 1/9) - 1 becomes 2x - 2x + 1/3 - 1. The terms 2x and -2x cancel each other out, leaving us with 1/3 - 1.Further Simplify: Further simplifying 1/3 - 1 gives us -2/3. So the right side of the equation simplifies to -2/3, which is equal to the left side, y', which we found to be -2/3.Verify Solution: Since both sides of the equation are equal, the function y=-(2)/(3)x+(1)/(9) satisfies the differential equation y'=2x+3y-1. Therefore, y=-(2)/(3)x+(1)/(9) is indeed a solution to the differential equation.

$y^{\prime}=2 x+3 y-1$ $\newline$ Is $y=-\frac{2}{3} x+\frac{1}{9}$ a solution to the above equation? $\newline$ Choose $1$ answer: $\newline$ (A) Yes $\newline$ (B) No

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$y^{\prime}=2 x+3 y-1$ $\newline$ Is $y=-\frac{2}{3} x+\frac{1}{9}$ a solution to the above equation? $\newline$ Choose $1$ answer: $\newline$ (A) Yes $\newline$ (B) No