Solve the equation. (dy)/(dx)=-(e^(x)y^(2))/(10) Choose 1 answer: (A) y=(10)/(e^(x))+C (B) y=(10)/(e^(x)+C) (C) y=(10)/(Ce^(x)) (D) y=(10+C)/(e^(x))

Separate variables: This is a first-order separable differential equation. To solve it, we need to separate the variables y and x on different sides of the equation. (dy)/(dx)=-(e^(x)y^(2))/(10) We can rewrite this as: (dy)/(y^2) = -(e^(x)/10)dxIntegrate both sides: Now we integrate both sides of the equation with respect to their variables. ∫(1/y^2)dy = -∫(e^(x)/10)dxApply constant of integration: The integral of 1/y^2 with respect to y is -1/y, and the integral of e^(x)/10 with respect to x is (1/10)e^(x). -1/y = -(1/10)e^(x) + C, where C is the constant of integration.Take reciprocal: We multiply both sides by -1 to get rid of the negative sign on the left side. 1/y = (1/10)e^(x) - CSimplify expression: Now we take the reciprocal of both sides to solve for y. y = 1/((1/10)e^(x) - C)Compare with options: To simplify the expression, we can multiply the numerator and denominator by 10. y = 10/(e^(x) - 10C) We can rename the constant -10C to a new constant C' for simplicity. y = 10/(e^(x) + C')We compare the final expression with the given options. The correct answer is (B) y=(10)/(e^(x)+C).

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Solve the equation.

(dy)/(dx)=-(e^(x)y^(2))/(10)
Choose 1 answer:
(A)
y=(10)/(e^(x))+C
(B)
y=(10)/(e^(x)+C)
(C)
y=(10)/(Ce^(x))
(D)
y=(10+C)/(e^(x))

Solve the equation. $\newline$ $\frac{d y}{d x}=-\frac{e^{x} y^{2}}{10}$ $\newline$ Choose $1$ answer: $\newline$ (A) $y=\frac{10}{e^{x}}+C$ $\newline$ (B) $y=\frac{10}{e^{x}+C}$ $\newline$ (C) $y=\frac{10}{C e^{x}}$ $\newline$ (D) $y=\frac{10+C}{e^{x}}$

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Calculus

Find derivatives of using multiple formulae

Full solution

Q. Solve the equation.

\newline

\frac{d y}{d x}=-\frac{e^{x} y^{2}}{10}

\newline

Choose

1

answer:

\newline

(A)

y=\frac{10}{e^{x}}+C

\newline

(B)

y=\frac{10}{e^{x}+C}

\newline

(C)

y=\frac{10}{C e^{x}}

\newline

(D)

y=\frac{10+C}{e^{x}}

Separate variables: This is a first-order separable differential equation. To solve it, we need to separate the variables $y$ and $x$ on different sides of the equation. $\newline$ $\frac{dy}{dx} = -\frac{e^{x}y^{2}}{10}$ $\newline$ We can rewrite this as: $\newline$ $\frac{dy}{y^2} = -\frac{e^{x}}{10}dx$
Integrate both sides: Now we integrate both sides of the equation with respect to their variables. $\newline$ $\int(\frac{1}{y^2})dy = -\int(\frac{e^{x}}{10})dx$
Apply constant of integration: The integral of $\frac{1}{y^2}$ with respect to $y$ is $-\frac{1}{y}$ , and the integral of $\frac{e^{x}}{10}$ with respect to $x$ is $\left(\frac{1}{10}\right)e^{x}$ . $\newline$ $-\frac{1}{y} = -\left(\frac{1}{10}\right)e^{x} + C$ , where $C$ is the constant of integration.
Take reciprocal: We multiply both sides by $-1$ to get rid of the negative sign on the left side. $\newline$ $\frac{1}{y} = \frac{1}{10}e^{x} - C$
Simplify expression: Now we take the reciprocal of both sides to solve for $y$ . $y = \frac{1}{\left(\frac{1}{10}e^{x} - C\right)}$
Compare with options: To simplify the expression, we can multiply the numerator and denominator by $10$ . $y = \frac{10}{e^{x} - 10C}$ We can rename the constant $-10C$ to a new constant $C'$ for simplicity. $y = \frac{10}{e^{x} + C'}$
Compare with options: To simplify the expression, we can multiply the numerator and denominator by $10$ . $\newline$ $y = \frac{10}{e^{x} - 10C}$ $\newline$ We can rename the constant $-10C$ to a new constant $C'$ for simplicity. $\newline$ $y = \frac{10}{e^{x} + C'}$ We compare the final expression with the given options. $\newline$ The correct answer is (B) $y=\frac{10}{e^{x}+C}$ .