Q. 40 Use antidifferentiation and/or s... [FREE SOLUTION]

Chapter 6: Q. 40 (page 570)

Use antidifferentiation and/or separation of variables to solve each of the initial-value problems in Exercises 29鈥�52
$\frac{d y}{d x} = \frac{2 x}{y^{2}}, y (0) = 4$

Short Answer

Expert verified

The solution of the differential equation and obtain the solution of the initial-value problem $\frac{d y}{d x} = \frac{2 x}{y^{3}} a s y (x) = \sqrt[3]{3 x^{2} + 64}$

Step by step solution

Step 1. Given information

The given initial value problem $\frac{d y}{d x} = \frac{2 x}{y^{2}}, y (0) = 4 . . . . . . . . . . . . . . . . (1)$

Step 2. Use antidifferentiation and/or separation of variables to solve each of the initial-value

Note that the differential equation involved in (1) is of the form

\frac{d y}{d x} = p (x) p (y)

in which

p (x) = 2 x

, and

q (y) = \frac{1}{y^{2}}

. So, the differential equation can be solved by applying variable separable method. Thus, the solution of the differential equation involved in the initial- value problem is given by

鈭� y^{2} d y = 鈭� 2 x d x \frac{1}{3} y^{3} = x^{2} + C_{1} y^{3} = 3 x^{2} + C (3 C_{1} = C) y = \sqrt[3]{3 x^{2} + C}

Now, use the given initial condition

y (0) = 4

, that is take

x = 0, y = 4

in the above result and evaluate the constant C.

$4 = \sqrt[3]{C} C = 64$

Substitute this value of the constant C in the solution of the differential equation and obtain the solution of the initial-value problem

\frac{d y}{d x} = \frac{2 x}{y^{3}} a s y (x) = \sqrt[3]{3 x^{2} + 64}

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Use antidifferentiation and/or separation of variables to solve each of the initial-value problems in Exercises 29鈥�52
$\frac{d y}{d x} = \frac{2 x}{y^{2}}, y (0) = 4$

Short Answer

Step by step solution

Step 1. Given information

Step 2. Use antidifferentiation and/or separation of variables to solve each of the initial-value

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91影视

Use antidifferentiation and/or separation of variables to solve each of the initial-value problems in Exercises 29鈥�52dydx=2xy2,y(0)=4

Short Answer

Step by step solution

Step 1. Given information

Step 2. Use antidifferentiation and/or separation of variables to solve each of the initial-value

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Probability and Statistics

Discrete Mathematics

Calculus

Pure Maths

Mechanics Maths

Study anywhere. Anytime. Across all devices.

Use antidifferentiation and/or separation of variables to solve each of the initial-value problems in Exercises 29鈥�52
$\frac{d y}{d x} = \frac{2 x}{y^{2}}, y (0) = 4$