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(dy)/(dx)=(5y)/(x)
Is
y=-2x^(5) a solution to the above equation?
Choose 1 answer:
(A) Yes
(B)
No

$\frac{d y}{d x}=\frac{5 y}{x}$ $\newline$ Is $y=-2 x^{5}$ a solution to the above equation? $\newline$ Choose $1$ answer: $\newline$ (A) Yes $\newline$ (B) $\mathrm{No}$

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

Full solution

Q.

\frac{d y}{d x}=\frac{5 y}{x}

\newline

Is

y=-2 x^{5}

a solution to the above equation?

\newline

Choose

1

answer:

\newline

(A) Yes

\newline

(B)

\mathrm{No}

Differentiate $y$ : To determine if $y = -2x^5$ is a solution to the differential equation $\frac{dy}{dx} = \frac{5y}{x}$ , we need to differentiate $y$ with respect to $x$ and then substitute the result into the differential equation to see if it holds true. $\newline$ Let's differentiate $y = -2x^5$ with respect to $x$ . $\newline$ $\frac{dy}{dx} = \frac{d(-2x^5)}{dx}$ $\newline$ $\frac{dy}{dx} = -2 \cdot 5x^{5-1}$ $\newline$ $\frac{dy}{dx} = -10x^4$
Substitute into equation: Now we substitute $y = -2x^5$ into the right side of the differential equation $\frac{5y}{x}$ to see if it equals the derivative we found. $\newline$ $\frac{5y}{x} = \frac{5(-2x^5)}{x}$ $\newline$ $\frac{5y}{x} = -10x^4$
Compare derivative and equation: We compare the derivative $\frac{dy}{dx} = -10x^4$ with the expression we found by substituting $y$ into the differential equation, which is also $-10x^4$ . Since both sides are equal, $y = -2x^5$ is indeed a solution to the differential equation $\frac{dy}{dx} = \frac{5y}{x}$ .

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