Chapter 2: Q 8RP (page 79)

Question: In Problems , solve the equation.
$\frac{dy}{dx} + \frac{2 y}{x} = 2 x^{2} y^{2}$

Short Answer

Expert verified

The solution of the given equation is $y = {(- 2 x^{3} + {Cx}^{2})}^{- 1}$ .

Step by step solution

Given information and simplification

Given that, $\frac{dy}{dx} + \frac{2 y}{x} = 2 x^{2} y^{2} \cdot \cdot \cdot \cdot \cdot \cdot (1)$

Multiply by y^-2 in equation (1),

$y^{- 2} \frac{dy}{dx} + \frac{2}{xy} = 2 x^{2} \cdot \cdot \cdot \cdot \cdot \cdot (2)$

Now substitute u = y^-1 differentiate with respect to x.

$\begin{matrix} \frac{du}{dx} = - y^{- 2} \frac{dy}{dx} \\ - \frac{du}{dx} = y^{- 2} \frac{dy}{dx} \end{matrix}$

Substitute in equation (2).

$\begin{matrix} - \frac{du}{dx} + \frac{2}{x} u = 2 x^{2} \\ \frac{du}{dx} - \frac{2}{x} u = - 2 x^{2} \cdot \cdot \cdot \cdot \cdot \cdot (3) \end{matrix}$

Let $P (x) = - \frac{2}{x}$

Find the value of $μ (x)$ .

$\begin{matrix} μ (x) = e^{\int P (x) dx} \\ = e^{- \int \frac{2}{x} dx} \\ = e^{- 2 \ln |x|} \\ = x^{- 2} \end{matrix}$

Evaluation method

Multiply x^-2 in equation (3) on both sides.

$\begin{matrix} x^{- 2} \frac{du}{dx} - x^{- 2} \frac{2}{x} u = - 2 \\ \frac{1}{x^{2}} \frac{du}{dx} - \frac{2}{x^{3}} u = - 2 \\ \frac{d}{dx} [\frac{1}{x^{2}} u] = - 2 \end{matrix}$

Now integrate the equation on both sides.

$\begin{matrix} \int \frac{d}{dx} [\frac{1}{x^{2}} u] dx = \int - 2 dx \\ \frac{1}{x^{2}} u = - 2 x + C \\ u = - 2 x^{3} + {Cx}^{2} \end{matrix}$

Substitute $u = y^{- 1}$ , we get,

$\begin{matrix} y^{- 1} = - 2 x^{3} + {Cx}^{2} \\ y = {(- 2 x^{3} + {Cx}^{2})}^{- 1} \end{matrix}$

So, the solution is $y = {(- 2 x^{3} + {Cx}^{2})}^{- 1}$

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Question: In Problems , solve the equation.
$\frac{dy}{dx} + \frac{2 y}{x} = 2 x^{2} y^{2}$

Short Answer

Step by step solution

Given information and simplification

Evaluation method

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Question: In Problems , solve the equation.dydx+2yx=2x2y2

Short Answer

Step by step solution

Given information and simplification

Evaluation method

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Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Pure Maths

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Study anywhere. Anytime. Across all devices.

Question: In Problems , solve the equation.
$\frac{dy}{dx} + \frac{2 y}{x} = 2 x^{2} y^{2}$