Q 4RP Question: In Problems 1-30, solv... [FREE SOLUTION]

91影视

Fundamentals Of Differential Equations And Boundary Value Problems

R. Kent Nagle, Edward B. Saff, Arthur David Snider

$Math Studyset 91影视 Explanations$ Math

9th Edition 路 616 pages

ISBN: 9780321977069

Chapter 2: Q 4RP (page 79)

Question: In Problems 1-30, solve the equation.
$鈥� \frac{dy}{dx} + \frac{3 y}{x} = x^{2} - 4 x + 3$

Short Answer

Expert verified

The solution of the given equation is $y = \frac{x^{3}}{6} - \frac{4 x^{2}}{5} + \frac{3 x}{4} + {Cx}^{- 3}$ .

Step by step solution

Given information and simplification

Given that, $\frac{dy}{dx} + \frac{3 y}{x} = x^{2} - 4 x + 3 鈰� 鈰� 鈰� 鈰� 鈰� 鈰� (1)$

Let $P (x) = \frac{3}{x}$ .

Find the value of $渭 (x)$ .

$\begin{matrix} 渭 (x) = e^{鈭� P (x) dx} \\ = e^{鈭� \frac{3}{x} dx} \\ = e^{3 \ln |x|} \\ = x^{3} \end{matrix}$

Multiply x³ in equation (1) on both sides.

$\begin{matrix} x^{3} \frac{dy}{dx} + x^{3} \frac{3 y}{x} = x^{3} (x^{2} - 4 x + 3) \\ x^{3} \frac{dy}{dx} + 3 x^{2} y = x^{5} - 4 x^{4} + 3 x^{3} \\ \frac{d}{dx} [x^{3} y] = x^{5} - 4 x^{4} + 3 x^{3} \end{matrix}$

Integration

Now integrate the equation on both sides.

$\begin{matrix} 鈭� \frac{d}{dx} [x^{3} y] dx = 鈭� (x^{5} - 4 x^{4} + 3 x^{3}) dx \\ x^{3} y = \frac{x^{6}}{6} - \frac{4 x^{5}}{5} + \frac{3 x^{4}}{4} + C \\ y = \frac{x^{- 3} x^{6}}{6} - \frac{4 x^{- 3} x^{5}}{5} + \frac{3 x^{- 3} x^{4}}{4} + {Cx}^{- 3} \\ y = \frac{x^{3}}{6} - \frac{4 x^{2}}{5} + \frac{3 x}{4} + {Cx}^{- 3} \end{matrix}$

So, the solution is $y = \frac{x^{3}}{6} - \frac{4 x^{2}}{5} + \frac{3 x}{4} + {Cx}^{- 3}$

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91影视

Question: In Problems 1-30, solve the equation.
$鈥� \frac{dy}{dx} + \frac{3 y}{x} = x^{2} - 4 x + 3$

Short Answer

Step by step solution

Given information and simplification

Integration

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91影视

Question: In Problems 1-30, solve the equation.鈥�dydx+3yx=x2-4x+3

Short Answer

Step by step solution

Given information and simplification

Integration

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

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Question: In Problems 1-30, solve the equation.
$鈥� \frac{dy}{dx} + \frac{3 y}{x} = x^{2} - 4 x + 3$