Solved on Feb 14, 2024

Solve the separable differential equation $\frac{dy}{dx} + 4xy^2 = 0$ .

STEP 1

Assumptions
1. We have a separable differential equation of the form $\frac{d y}{d x} + 4xy^2 = 0$ .
2. We need to separate the variables $x$ and $y$ to integrate them individually.

STEP 2

Rewrite the differential equation to isolate the terms involving $y$ on one side and the terms involving $x$ on the other side.
$\frac{d y}{d x} = -4xy^2$

STEP 3

Separate the variables by dividing both sides by $y^2$ and multiplying both sides by $dx$ .
$\frac{1}{y^2} dy = -4x dx$

STEP 4

Integrate both sides of the equation with respect to their respective variables.
$\int \frac{1}{y^2} dy = \int -4x dx$

STEP 5

Calculate the integral on the left side.
$\int \frac{1}{y^2} dy = \int y^{-2} dy = -y^{-1} + C_1$

STEP 6

Calculate the integral on the right side.
$\int -4x dx = -2x^2 + C_2$

STEP 7

Combine the constants of integration $C_1$ and $C_2$ into a single constant $C$ since only the relative difference between them matters. We can write the integrated equation as:
$-y^{-1} = -2x^2 + C$

STEP 8

Multiply both sides of the equation by $-1$ to simplify.
$y^{-1} = 2x^2 - C$

STEP 9

Take the reciprocal of both sides to solve for $y$ .
$y = \frac{1}{2x^2 - C}$
The general solution to the differential equation is $y = \frac{1}{2x^2 - C}$ , where $C$ is the constant of integration.

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Solve the separable differential equation $\frac{dy}{dx} + 4xy^2 = 0$ .

STEP 1

Assumptions
1. We have a separable differential equation of the form $\frac{d y}{d x} + 4xy^2 = 0$ .
2. We need to separate the variables $x$ and $y$ to integrate them individually.

STEP 2

Rewrite the differential equation to isolate the terms involving $y$ on one side and the terms involving $x$ on the other side.
$\frac{d y}{d x} = -4xy^2$

STEP 3

Separate the variables by dividing both sides by $y^2$ and multiplying both sides by $dx$ .
$\frac{1}{y^2} dy = -4x dx$

STEP 4

Integrate both sides of the equation with respect to their respective variables.
$\int \frac{1}{y^2} dy = \int -4x dx$

STEP 5

Calculate the integral on the left side.
$\int \frac{1}{y^2} dy = \int y^{-2} dy = -y^{-1} + C_1$

STEP 6

Calculate the integral on the right side.
$\int -4x dx = -2x^2 + C_2$

STEP 7

Combine the constants of integration $C_1$ and $C_2$ into a single constant $C$ since only the relative difference between them matters. We can write the integrated equation as:
$-y^{-1} = -2x^2 + C$

STEP 8

Multiply both sides of the equation by $-1$ to simplify.
$y^{-1} = 2x^2 - C$

STEP 9

Take the reciprocal of both sides to solve for $y$ .
$y = \frac{1}{2x^2 - C}$
The general solution to the differential equation is $y = \frac{1}{2x^2 - C}$ , where $C$ is the constant of integration.

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Solve the separable differential equation dydx+4xy2=0\frac{dy}{dx} + 4xy^2 = 0dxdy​+4xy2=0.

STEP 1

Assumptions1. We have a separable differential equation of the form dydx+4xy2=0\frac{d y}{d x} + 4xy^2 = 0dxdy​+4xy2=0.2. We need to separate the variables xxx and yyy to integrate them individually.

STEP 2

Rewrite the differential equation to isolate the terms involving yyy on one side and the terms involving xxx on the other side.dydx=−4xy2\frac{d y}{d x} = -4xy^2dxdy​=−4xy2

STEP 3

Separate the variables by dividing both sides by y2y^2y2 and multiplying both sides by dxdxdx.1y2dy=−4xdx\frac{1}{y^2} dy = -4x dxy21​dy=−4xdx

STEP 4

Integrate both sides of the equation with respect to their respective variables.∫1y2dy=∫−4xdx\int \frac{1}{y^2} dy = \int -4x dx∫y21​dy=∫−4xdx

STEP 5

Calculate the integral on the left side.∫1y2dy=∫y−2dy=−y−1+C1\int \frac{1}{y^2} dy = \int y^{-2} dy = -y^{-1} + C_1∫y21​dy=∫y−2dy=−y−1+C1​

STEP 6

Calculate the integral on the right side.∫−4xdx=−2x2+C2\int -4x dx = -2x^2 + C_2∫−4xdx=−2x2+C2​

STEP 7

Combine the constants of integration C1C_1C1​ and C2C_2C2​ into a single constant CCC since only the relative difference between them matters. We can write the integrated equation as:−y−1=−2x2+C-y^{-1} = -2x^2 + C−y−1=−2x2+C

STEP 8

Multiply both sides of the equation by −1-1−1 to simplify.y−1=2x2−Cy^{-1} = 2x^2 - Cy−1=2x2−C

STEP 9

Take the reciprocal of both sides to solve for yyy.y=12x2−Cy = \frac{1}{2x^2 - C}y=2x2−C1​The general solution to the differential equation is y=12x2−Cy = \frac{1}{2x^2 - C}y=2x2−C1​, where CCC is the constant of integration.

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Solve the separable differential equation $\frac{dy}{dx} + 4xy^2 = 0$ .

Assumptions
1. We have a separable differential equation of the form $\frac{d y}{d x} + 4xy^2 = 0$ .
2. We need to separate the variables $x$ and $y$ to integrate them individually.

Rewrite the differential equation to isolate the terms involving $y$ on one side and the terms involving $x$ on the other side.
$\frac{d y}{d x} = -4xy^2$

Separate the variables by dividing both sides by $y^2$ and multiplying both sides by $dx$ .
$\frac{1}{y^2} dy = -4x dx$

Integrate both sides of the equation with respect to their respective variables.
$\int \frac{1}{y^2} dy = \int -4x dx$

Calculate the integral on the left side.
$\int \frac{1}{y^2} dy = \int y^{-2} dy = -y^{-1} + C_1$

Calculate the integral on the right side.
$\int -4x dx = -2x^2 + C_2$

Combine the constants of integration $C_1$ and $C_2$ into a single constant $C$ since only the relative difference between them matters. We can write the integrated equation as:
$-y^{-1} = -2x^2 + C$

Multiply both sides of the equation by $-1$ to simplify.
$y^{-1} = 2x^2 - C$

Take the reciprocal of both sides to solve for $y$ .
$y = \frac{1}{2x^2 - C}$
The general solution to the differential equation is $y = \frac{1}{2x^2 - C}$ , where $C$ is the constant of integration.