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

Separating Variables: Anastasia starts by separating variables. (dy)/(dx) = y^(2)sin(x) To separate variables, we need to get all y terms on one side and all x terms on the other side. (dy)/y^(2) = sin(x)dx This is the correct separation of variables.Integrating Both Sides: Anastasia integrates both sides. ∫(1/y^(2))dy = ∫sin(x)dx The integral of 1/y^2 with respect to y is -1/y, and the integral of sin(x) with respect to x is -cos(x). So, -1/y = -cos(x) + C₁ Anastasia's integration result is incorrect.

Resources

Testimonials

Plans

Resources

Testimonials

Plans

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

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

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

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

View step-by-step help

Home

Math Problems

Algebra 1

Compare linear and exponential growth

Full solution

Q. Anastasia tried to solve the differential equation

\frac{d y}{d x}=y^{2} \sin (x)

. This is her work:

\newline

\frac{d y}{d x}=y^{2} \sin (x)

\newline

Step

1

\quad \int y^{2} d y=\int \sin (x) d x

\newline

Step

2

\quad \frac{y^{3}}{3}=-\cos (x)+C_{1}

\newline

Step

3

\quad y^{3}=-3 \cos (x)+C

\newline

Step

4

\quad y=\sqrt[3]{-3 \cos (x)+C}

\newline

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

\newline

Choose

1

answer:

\newline

(A) Anastasia's work is correct.

\newline

(B) Step

1

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

\newline

2

is incorrect. Anastasia didn't integrate

\sin (x)

correctly.

\newline

(D) Step

4

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

\sqrt[3]{-\cos (x)}+C

Separating Variables: Anastasia starts by separating variables. $\newline$ $(\frac{dy}{dx}) = y^{2}\sin(x)$ $\newline$ To separate variables, we need to get all $y$ terms on one side and all $x$ terms on the other side. $\newline$ $(\frac{dy}{y^{2}}) = \sin(x)dx$ $\newline$ This is the correct separation of variables.
Integrating Both Sides: Anastasia integrates both sides. $\newline$ $\int(1/y^{2})dy = \int\sin(x)dx$ $\newline$ The integral of $1/y^2$ with respect to $y$ is $-1/y$ , and the integral of $\sin(x)$ with respect to $x$ is $-\cos(x)$ . $\newline$ So, $-1/y = -\cos(x) + C_1$ $\newline$ Anastasia's integration result is incorrect.