Chapter 6: Q. 6 (page 569)

Verify that $y (x) = \sqrt{x + 9}$ is a solution of the initial-value problem $\frac{d y}{d x} = \frac{1}{2 y}$ and $y (0) = 3$ .

Short Answer

Expert verified

The solution of the given differential equation has been verified.

Step by step solution

Step 1. Given information:

The differential equation: $\frac{d y}{d x} = \frac{1}{2 y}$

$y (0) = 3$

The solution of this differential equation $y (x) = \sqrt{x + 9}$

Step 2. Verify the solution.

$\frac{d y}{d x} = \frac{1}{2 y} 2 y d y = d x$

Now integrate both sides.

$\int 2 y d y = \int d x 2 \int y d y = \int d x 2 . \frac{y^{2}}{2} = x + c y^{2} = x + c y = \sqrt{x + c}$

Now find $c$ by substituting $y (0) = 3$

$3 = \sqrt{0 + c} 3 = \sqrt{c} c = 9$

So, the solution is: $y (x) = \sqrt{x + 9}$

Thus, the solution has been verified.

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.

Verify that $y (x) = \sqrt{x + 9}$ is a solution of the initial-value problem $\frac{d y}{d x} = \frac{1}{2 y}$ and $y (0) = 3$ .

Short Answer

Step by step solution

Step 1. Given information:

Step 2. Verify the solution.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Applied Mathematics

Theoretical and Mathematical Physics

Calculus

Discrete Mathematics

Pure Maths

Study anywhere. Anytime. Across all devices.

Verify that y(x)=x+9is a solution of the initial-value problem dydx=12yand y(0)=3.

Short Answer

Step by step solution

Step 1. Given information:

Step 2. Verify the solution.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Applied Mathematics

Theoretical and Mathematical Physics

Calculus

Discrete Mathematics

Pure Maths

Study anywhere. Anytime. Across all devices.

Verify that $y (x) = \sqrt{x + 9}$ is a solution of the initial-value problem $\frac{d y}{d x} = \frac{1}{2 y}$ and $y (0) = 3$ .