Chapter 6: Q. 20 (page 570)

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

Short Answer

Expert verified

Ans: The solution to the differential equation $\frac{d y}{d x} = \frac{x + 1}{\sqrt{x}}$ is $\frac{2}{3} x^{3 / 2} + 2 \sqrt{x} + C$

Step by step solution

Step 1. Given information.

given,

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

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.

$\frac{d y}{d x} = \frac{x + 1}{\sqrt{x}} \dots \dots (1)$

Step 3. Now,

Note that the differential equation $(1)$ does not contain the dependent variable at all, so technically the variables have already been separated. So, the differential equation can be solved by antidifferentiation. Thus, the solution of the differential equation is obtained by integrating both the sides

localid="1649151533535" $\begin{aligned} \int d y = \int \frac{x + 1}{\sqrt{x}} d x \\ = \int (\sqrt{x} + \frac{1}{\sqrt{x}}) d x \\ = \int \sqrt{x} d x + \int x^{- 1 / 2} d x \\ = \frac{2}{3} x^{3 / 2} + 2 \sqrt{x} + C \end{aligned}$

Hence, a solution to the differential equation $\frac{d y}{d x} = \frac{x + 1}{\sqrt{x}}$ is localid="1649151637606" $\frac{2}{3} x^{3 / 2} + 2 \sqrt{x} + C$ .

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

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. Now,

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

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. Now,

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Statistics

Theoretical and Mathematical Physics

Pure Maths

Mechanics Maths

Discrete Mathematics

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.
$\frac{d y}{d x} = \frac{x + 1}{\sqrt{x}}$