Chapter 6: Q. 22 (page 570)

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

Short Answer

Expert verified

Ans: The solution of the differential equation $\frac{d y}{d x} = - 3 x y$ is $y = A e^{- \frac{3}{2} x^{2}}$ .

Step by step solution

Step 1. Given information.

given,

$\frac{d y}{d x} = - 3 x y$

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.

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

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

$\begin{aligned} \int \frac{1}{y} d y = - \int 3 x d x \\ \ln | y | = - \frac{3}{2} x^{2} + C \\ y = e^{- \frac{3}{2} x^{2} + C} \\ = A e^{- \frac{3}{2} x^{2}} \end{aligned}$

Hence a solution to the differential equation $\frac{d y}{d x} = - 3 x y$ is localid="1649166959230" $y = A e^{- \frac{3}{2} x^{2}}$

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

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

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

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Decision Maths

Theoretical and Mathematical Physics

Discrete Mathematics

Logic and Functions

Mechanics Maths

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. dydx=−3xy

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

Geometry

Decision Maths

Theoretical and Mathematical Physics

Discrete Mathematics

Logic and Functions

Mechanics Maths

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.
$\frac{d y}{d x} = - 3 x y$