Chapter 6: Q. 25 (page 570)

Use antidifferentiation and/or separation of variables to solve the given differential equations. Your answers will involve unsolved constants.
$\frac{d y}{d x} = \frac{3 x + 1}{x y}$

Short Answer

Expert verified

Ans: The solution of the differential equation $\frac{d y}{d x} = \frac{3 x + 1}{x y}$ is $y = \pm \sqrt{6 x + 2 \ln | x | + C}$

Step by step solution

Step 1. Given information.

given,

$\frac{d y}{d x} = \frac{3 x + 1}{x y}$

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.

$\frac{d y}{d x} = \frac{3 x + 1}{x y} . . . . . . (1)$

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 $p (x) = \frac{3 x + 1}{x}$ and $q (y) = \frac{1}{y}$ . So the differential equation can be solved by applying the variable separable method. Separate the variables and integrate both the sides

$\begin{aligned} \int y d y = \int \frac{3 x + 1}{x} d x \\ \frac{y^{2}}{2} = \int 3 d x + \int \frac{1}{x} d x \\ = 3 x + \ln | x | + C_{1} \\ y^{2} = 6 x + 2 \ln | x | + 2 C_{1} \end{aligned}$

Replace the constant localid="1649177740460" $2 C_{1}$ by another constant $C$ and write the solution as $y^{2} = 6 x + 2 \ln | x | + C$

Hence a solution to the differential equation $\frac{d y}{d x} = \frac{3 x + 1}{x y}$ is localid="1649177691002" $y = \pm \sqrt{6 x + 2 \ln | x | + C}$ .

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Use antidifferentiation and/or separation of variables to solve the given differential equations. Your answers will involve unsolved constants.
$\frac{d y}{d x} = \frac{3 x + 1}{x y}$

Short Answer

Step by step solution

Step 1. Given information.

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.

Step 3. Solution

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Use antidifferentiation and/or separation of variables to solve the given differential equations. Your answers will involve unsolved constants. dydx=3x+1xy

Short Answer

Step by step solution

Step 1. Given information.

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.

Step 3. Solution

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Decision Maths

Pure Maths

Logic and Functions

Mechanics Maths

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

Use antidifferentiation and/or separation of variables to solve the given differential equations. Your answers will involve unsolved constants.
$\frac{d y}{d x} = \frac{3 x + 1}{x y}$