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Chapter 6: Q. 24 (page 570)

Use antidifferentiation and/or separation of variables to solve the given differential equations. Your answers will involve unsolved constants.
$\frac{d y}{d x} = x y^{2}$

Short Answer

Expert verified

Ans: The solution of the differential equation $\frac{d y}{d x} = x y^{2}$ is $y = - \frac{2}{x^{2} + C}$ .

Step by step solution

01

Step 1. Given information.

given,

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

02

Step 2. Consider the differential equation defined by equation (1) given below and solve it by using antidifferentiation and/or separation of the variable method.

$\frac{d y}{d x} = x y^{2} . . . . . (1)$

03

Step 3. Solution

Note that the differential equation (1) is of the form of $\frac{d y}{d x} = p (x) q (y)$ in which $p (x) = x$ and $q (y) = y^{2}$ . So the differential equation can be solved by applying the variable separable method. Separate the variables and integrate both the sides

$\begin{aligned} 鈭� \frac{1}{y^{2}} d y = 鈭� x d x \\ 鈭� \frac{1}{y} = \frac{1}{2} x^{2} + C_{1} \\ y = 鈭� \frac{2}{x^{2} + 2 C_{1}} \\ = 鈭� \frac{2}{x^{2} + C} (2 C_{1} = C) \end{aligned}$

Hence a solution to the differential equation

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

is

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

.

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Most popular questions from this chapter

For each pair of definite integrals in exercise 13-18 decide which if either is larger without computing the integrals

$蟺 {鈭�}_{0}^{\frac{蟺}{2}} \cos^{2} (x) d x a n d 蟺 {鈭�}_{\frac{蟺}{2}}^{蟺} \cos^{2} (x) d x$

$\frac{d y}{d x} = 5 y, y (0) = 1$

Consider the region between the graph of $f (x) = x - 2$ and the x-axis on [2, 5]. For each line of rotation given in Exercises 35鈥�40, use definite integrals to find the volume of the resulting solid.

Consider the region between the graph of $f (x) = x - 2$ and the x-axis on [2, 5]. For each line of rotation given in Exercises 35鈥�40, use definite integrals to find the volume of the resulting solid.

Use antidifferentiation and/or separation of variables to solve the given differential equations. Your answers will involve unsolved constants.

$\frac{d y}{d x} = y \sin 鈦� x$

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