Chapter 1: Q8-3-14P (page 1)

Using (3.9), find the general solution of each of the following differential equations. Compare a computer solution and, if necessary, reconcile it with yours. Hint: See comments just after()3.9, and Example 1.
14. $\frac{dy}{dx} = \frac{3y}{{3y}^{2/3} - x}$
Hint: For Problems 12to 14, solve forx in terms of y.

Short Answer

Expert verified

$x = y^{2/3} + {Cy}^{- 1/3}$

Step by step solution

Meaning of first-order differentiation

The linear differential equation is defined by $x' + Px = Q$ , where Pand Qare numeric constants or functions in x. It is made up of ay and a yderivative. The differential equation is called the first-order linear differential equation because it is a first-order differentiation.

Given Parameters

Given an equation $\frac{dy}{dx} = \frac{3y}{{3y}^{2/3} - x}$

Write the differential equation in x'+Px=Q

Writing the given differential equation in the form $x' + Px = Qx'$ :

$x^{'} = \frac{{3y}^{2/3} - x}{3y} x^{'} + \frac{x}{3y} = \frac{1}{y^{1/3}}$

From eq.3.4 ,

$I = \int \frac{dy}{3y} = \frac{lny}{3} e^{I} = e^{\frac{lny}{3}} = y^{1/3}$

Find the general solution of the given differential equation

The solution will be as follows

${xe}^{I} = \int (\frac{1}{y^{1/3}}) y^{1/3} dy = y + C x = y^{2/3} + {Cy}^{- 1/3}$

So, the general solution is $x = y^{2/3} + {Cy}^{- 1/3}$

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Short Answer

Step by step solution

Meaning of first-order differentiation

Given Parameters

Write the differential equation in x'+Px=Q

Find the general solution of the given differential equation

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Using (3.9), find the general solution of each of the following differential equations. Compare a computer solution and, if necessary, reconcile it with yours. Hint: See comments just after()3.9, and Example 1.14.dydx=3y3y2/3-xHint: For Problems 12to 14, solve forx in terms of y.

Short Answer

Step by step solution

Meaning of first-order differentiation

Given Parameters

Write the differential equation in x'+Px=Q

Find the general solution of the given differential equation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Electromagnetism

Measurements

Nuclear Physics

Torque and Rotational Motion

Magnetism and Electromagnetic Induction

Fundamentals of Physics

Study anywhere. Anytime. Across all devices.