Solve the equation. (dy)/(dx)=4ye^(x) Choose 1 answer: (A) y=e^(Ce^(x)) (B) y=e^(e^(x)+C) (C) y=e^(4e^(x))+C (D) y=Ce^(4e^(x))

Recognize the equation type: Recognize that the given differential equation (dy)/(dx)=4ye^(x) is a first-order linear differential equation that can be solved using the method of separation of variables.Separate variables: Separate the variables by dividing both sides by y and multiplying both sides by dx to get (1/y)dy = 4e^(x)dx.Integrate both sides: Integrate both sides of the equation. The left side with respect to y and the right side with respect to x. ∫(1/y)dy = ∫4e^(x)dxPerform integration: Perform the integration on both sides. The integral of (1/y)dy is ln|y|, and the integral of 4e^(x)dx is 4e^(x). ln|y| = 4e^(x) + C, where C is the constant of integration.Solve for y: Solve for y by exponentiating both sides to eliminate the natural logarithm. e^(ln|y|) = e^(4e^(x) + C) |y| = e^(4e^(x)) * e^(C)Exponentiate both sides: Since e^(C) is just another constant, we can rename it as C' (C prime) for simplicity. |y| = C'e^(4e^(x))Remove absolute value: Remove the absolute value by considering that y can be either positive or negative, so y = ±C'e^(4e^(x)). However, since C' can absorb the negative sign, we can write y = C'e^(4e^(x)).Match with answer choices: Match the solution to the given answer choices. The correct answer choice that matches y = C'e^(4e^(x)) is (D) y=Ce^(4e^(x)).

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Solve the equation.

(dy)/(dx)=4ye^(x)
Choose 1 answer:
(A)
y=e^(Ce^(x))
(B)
y=e^(e^(x)+C)
(C)
y=e^(4e^(x))+C
(D)
y=Ce^(4e^(x))

Solve the equation. $\newline$ $\frac{d y}{d x}=4 y e^{x}$ $\newline$ Choose $1$ answer: $\newline$ (A) $y=e^{C e^{x}}$ $\newline$ (B) $y=e^{e^{x}+C}$ $\newline$ (C) $y=e^{4 e^{x}}+C$ $\newline$ (D) $y=C e^{4 e^{x}}$

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Q. Solve the equation.

\newline

\frac{d y}{d x}=4 y e^{x}

\newline

Choose

1

answer:

\newline

(A)

y=e^{C e^{x}}

\newline

(B)

y=e^{e^{x}+C}

\newline

(C)

y=e^{4 e^{x}}+C

\newline

(D)

y=C e^{4 e^{x}}

Recognize the equation type: Recognize that the given differential equation $\frac{dy}{dx}=4ye^{x}$ is a first-order linear differential equation that can be solved using the method of separation of variables.
Separate variables: Separate the variables by dividing both sides by $y$ and multiplying both sides by $dx$ to get $\frac{1}{y}$ $dy$ = $4e^{x}$ $dx$ .
Integrate both sides: Integrate both sides of the equation. The left side with respect to $y$ and the right side with respect to $x$ . $\int(\frac{1}{y})dy = \int 4e^{(x)}dx$
Perform integration: Perform the integration on both sides. $\newline$ The integral of $\frac{1}{y}$ $dy$ is $\ln|y|$ , and the integral of $4e^{x}$ $dx$ is $4e^{x}$ . $\newline$ $\ln|y| = 4e^{x} + C$ , where $C$ is the constant of integration.
Solve for $y$ : Solve for $y$ by exponentiating both sides to eliminate the natural logarithm. $\newline$ $e^{(\ln|y|)} = e^{(4e^{(x)} + C)}$ $\newline$ $|y| = e^{(4e^{(x)})} \cdot e^{(C)}$
Exponentiate both sides: Since $e^{C}$ is just another constant, we can rename it as $C'$ (C prime) for simplicity. $|y| = C'e^{4e^{x}}$
Remove absolute value: Remove the absolute value by considering that $y$ can be either positive or negative, so $y = \pm C'e^{4e^{x}}$ . However, since $C'$ can absorb the negative sign, we can write $y = C'e^{4e^{x}}$ .
Match with answer choices: Match the solution to the given answer choices. $\newline$ The correct answer choice that matches $y = C'e^{4e^{x}}$ is (D) $y=Ce^{4e^{x}}$ .