Baylee tried to solve the differential equation (dy)/(dx)=(e^(x))/(y^(2)). This is her work: (dy)/(dx)=(e^(x))/(y^(2)) Step 1: quad inty^(2)dy=inte^(x)dx Step 2: quad(y^(3))/(3)=e^(x) Step 3: quady^(3)=3e^(x) Step 4: quad y=root(3)(3e^(x))+C Is Baylee's work correct? If not, what is her mistake? Choose 1 answer: (A) Baylee's work is correct. (B) Step 1 is incorrect. The separation of variables wasn't done correctly. (C) Step 2 is incorrect. The right-hand side of the equation should be e^(x)+C. (D) Step 4 is incorrect. The right-hand side of the equation should be root(3)(3e^(x))+root(3)(C).

Check First Step: Baylee is trying to solve the differential equation by separating variables. Let's check her first step: (dy)/(dx) = (e^(x))/(y^(2)) To separate variables, we multiply both sides by y^2 and dx: y^2 dy = e^x dx This is the correct separation of variables.Separate Variables: Now, Baylee integrates both sides of the equation: ∫y^2 dy = ∫e^x dx The integral of y^2 with respect to y is (y^3)/3, and the integral of e^x with respect to x is e^x plus a constant of integration. (y^3)/3 = e^x + C Baylee forgot to include the constant of integration on the right-hand side.

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Baylee tried to solve the differential equation
(dy)/(dx)=(e^(x))/(y^(2)). This is her work:

(dy)/(dx)=(e^(x))/(y^(2))
Step 1:
quad inty^(2)dy=inte^(x)dx
Step 2:
quad(y^(3))/(3)=e^(x)
Step 3:
quady^(3)=3e^(x)
Step 4:
quad y=root(3)(3e^(x))+C
Is Baylee's work correct? If not, what is her mistake?
Choose 1 answer:
(A) Baylee's work is correct.
(B) Step 1 is incorrect. The separation of variables wasn't done correctly.
(C) Step 2 is incorrect. The right-hand side of the equation should be
e^(x)+C.
(D) Step 4 is incorrect. The right-hand side of the equation should be
root(3)(3e^(x))+root(3)(C).

Baylee tried to solve the differential equation $\frac{d y}{d x}=\frac{e^{x}}{y^{2}}$ . This is her work: $\newline$ $\frac{d y}{d x}=\frac{e^{x}}{y^{2}}$ $\newline$ Step $1$ : $\quad \int y^{2} d y=\int e^{x} d x$ $\newline$ Step $2$ : $\quad \frac{y^{3}}{3}=e^{x}$ $\newline$ Step $3$ : $\quad y^{3}=3 e^{x}$ $\newline$ Step $4$ : $\quad y=\sqrt[3]{3 e^{x}}+C$ $\newline$ Is Baylee's work correct? If not, what is her mistake? $\newline$ Choose $1$ answer: $\newline$ (A) Baylee's work is correct. $\newline$ (B) Step $1$ is incorrect. The separation of variables wasn't done correctly. $\newline$ (C) Step $2$ is incorrect. The right-hand side of the equation should be $e^{x}+C$ . $\newline$ (D) Step $4$ is incorrect. The right-hand side of the equation should be $\sqrt[3]{3 e^{x}}+\sqrt[3]{C}$ .

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Q. Baylee tried to solve the differential equation

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

. This is her work:

\newline

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

\newline

Step

1

\quad \int y^{2} d y=\int e^{x} d x

\newline

Step

2

\quad \frac{y^{3}}{3}=e^{x}

\newline

Step

3

\quad y^{3}=3 e^{x}

\newline

Step

4

\quad y=\sqrt[3]{3 e^{x}}+C

\newline

Is Baylee's work correct? If not, what is her mistake?

\newline

Choose

1

answer:

\newline

(A) Baylee's work is correct.

\newline

(B) Step

1

is incorrect. The separation of variables wasn't done correctly.

\newline

2

is incorrect. The right-hand side of the equation should be

e^{x}+C

\newline

(D) Step

4

is incorrect. The right-hand side of the equation should be

\sqrt[3]{3 e^{x}}+\sqrt[3]{C}

Check First Step: Baylee is trying to solve the differential equation by separating variables. Let's check her first step: $\newline$ $\frac{dy}{dx} = \frac{e^{x}}{y^{2}}$ $\newline$ To separate variables, we multiply both sides by $y^2$ and $dx$ : $\newline$ $y^2 dy = e^x dx$ $\newline$ This is the correct separation of variables.
Separate Variables: Now, Baylee integrates both sides of the equation: $\newline$ $\int y^2 \, dy = \int e^x \, dx$ $\newline$ The integral of $y^2$ with respect to $y$ is $(y^3)/3$ , and the integral of $e^x$ with respect to $x$ is $e^x$ plus a constant of integration. $\newline$ $(y^3)/3 = e^x + C$ $\newline$ Baylee forgot to include the constant of integration on the right-hand side.