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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 15RP (page 79)

Question: In Problems , solve the equation.
$\frac{dy}{dx} = 2 - \sqrt{2 x - y + 3}$

Short Answer

Expert verified

The solution of the given equation is $y = 2 x + 3 - \frac{{(x + C)}^{2}}{4}$ .

Step by step solution

01

Given information and simplification

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

Evaluate the equation (1).

Substitute u = 2x - y and $\frac{dy}{dx} = 2 - \frac{du}{dx}$ .

role="math" localid="1663945257707" $\begin{matrix} \frac{dy}{dx} = 2 - \sqrt{2 x - y + 3} \\ 2 - \frac{du}{dx} = 2 - \sqrt{u + 3} \\ \frac{du}{dx} = \sqrt{u + 3} \\ \frac{1}{\sqrt{u + 3}} du = dx \end{matrix} \frac{1}{\sqrt{u + 3}} du = dx 鈰� 鈰� 鈰� 鈰� 鈰� 鈰� (2)$

Now integrate the equation (2) on both sides.

$鈭� \frac{1}{\sqrt{u + 3}} du = 鈭� dx$

02

Evaluation method

Find the value.

$\begin{matrix} 鈭� \frac{1}{\sqrt{u + 3}} du = 鈭� dx \\ 2 \sqrt{u + 3} = x + C \\ \sqrt{u + 3} = \frac{x + C}{2} \\ u + 3 = {(\frac{x + C}{2})}^{2} \end{matrix}$

Substitute the value of u here.

$\begin{matrix} 2 x - y + 3 = \frac{{(x + C)}^{2}}{4} \\ - y = - 2 x - 3 + \frac{{(x + C)}^{2}}{4} \\ y = 2 x + 3 - \frac{{(x + C)}^{2}}{4} \end{matrix}$

So, the solution is $y = 2 x + 3 - \frac{{(x + C)}^{2}}{4}$

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Most popular questions from this chapter

Question: In Problems , solve the equation.

$\frac{dy}{dx} = {(x + y + 1)}^{2} - {(x + y - 1)}^{2}$

Question: Use the substitution $v = x - y + 2$ to solve equation (8).

$\frac{dy}{dx} = y - x - 1 + {(x - y + 2)}^{- 1}$

In problem $1 - 6$ , determine whether the differential equation is separable $\frac{dy}{dx} - s i n (x + y) = 0$ .

In problem $1 - 6$ , determine whether the differential equation is separablerole="math" localid="1654775979001" $(x y^{2} + 3 y^{2}) d y - 2 x d x = 0$ .

Question: In Problems 1-30, solve the equation.

$鈥� [\sin (xy) + x 鈥� y 鈥� \cos (xy)] dx + [1 + x^{2} \cos (xy)] dy = 0$

See all solutions

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