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12. If

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

and if

y=-1

when

x=1

, then when

x=2, y=

...

Jan 30, 2024

12. If

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

and if

y=-1

when

x=1

, then when

x=2, y=

(A)

-\frac{2}{3}

(B)

-\frac{1}{3}

\frac{1}{3}

(E)

\frac{2}{3}

Solution by Steps

step 2

Separating variables gives

\frac{dy}{y^2} = 2dx

. Integrating both sides, we get

-\frac{1}{y} = 2x + C

step 3

Using the initial condition

y(1) = -1

, we find

C = -2 - \frac{1}{-1} = -1

. So the particular solution is

-\frac{1}{y} = 2x - 1

step 4

Solving for

y

when

x = 2

, we substitute

2

into the equation to get

-\frac{1}{y} = 2(2) - 1 = 3

, which gives

y = -\frac{1}{3}

Key Concept

Separation of Variables and Integration

Explanation

To solve a first-order separable differential equation, we separate the variables and integrate both sides. The initial condition is then used to find the particular solution.