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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 11RP (page 79)

Question: In Problems 1 - 30, solve the equation.
$鈥� \frac{dx}{dt} = 1 + \cos^{2} (t - x)$

Short Answer

Expert verified

The solution of the given equation is $\tan (t - x) + t = C$ .

Step by step solution

01

Given information and simplification

Given that, $\frac{dx}{dt} = 1 + \cos^{2} (t - x) 鈰� 鈰� 鈰� 鈰� 鈰� 鈰� (1)$

Evaluate the equation (1).

Let us take $u = t - x$ . Then, $\frac{dx}{dt} = 1 - \frac{du}{dt}$ .

$\begin{matrix} \frac{dx}{dt} = 1 + \cos^{2} (t - x) \\ 1 - \frac{du}{dt} = 1 + \cos^{2} u \\ - \frac{du}{dt} = \cos^{2} u \\ - \frac{1}{\cos^{2} u} du = dt \end{matrix} - \frac{1}{\cos^{2} u} du = dt 鈰� 鈰� 鈰� 鈰� 鈰� 鈰� (2)$

02

Integration

Now integrate the equation (2) on both sides.

$\begin{matrix} - 鈭� \frac{1}{\cos^{2} u} du = 鈭� dt \\ - 鈭� \sec^{2} u 鈥� du = t + C \\ - \tan 鈥� u = t + C \end{matrix}$

Substitute the value of u here.

$\tan (t - x) + t = C$

So, the solution is $\tan (t - x) + t = C$

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Most popular questions from this chapter

In problems, 1-8 identify (do not solve) the equation as homogeneous, Bernoulli, linear coefficients, or of the form $y^{'} = G (ax + by)$ . $\frac{dy}{dx} + \frac{y}{x} = x^{3} y^{2}$

In problem $1 - 6$ , determine whether the given differential equation is separable $s^{2} + \frac{d s}{d t} = \frac{s + 1}{s t}$ .

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

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

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

$\frac{dy}{dx} - \frac{y}{x} = x^{2} \sin 鈥� 2 x$

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