Book edition 9th

Author(s) R. Kent Nagle, Edward B. Saff, Arthur David Snider

Pages 616 pages

ISBN 9780321977069

Answers without the blur.

Just sign up for free and you're in.

Short Answer

In problem 7-16, solve the equation. $\frac{dx}{dt} = \frac{t}{x e^{t + 2 x}}$

The solution of the given differential equation is $e^{2 x} (2 x - 1) + 4 e^{- t} (t + 1) = C$ .

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Concept of Separable Differential Equation

A first-order ordinary differential equation $\frac{dy}{dx} = f (x, y)$ is referred to as separable if the function in the right-hand side of the equation is expressed as a product of two functions $g (x)$ that is a function of x alone and $h (y)$ that is a function of y alone.

A separable differential equation can be expressed as $\frac{dy}{dx} = g (x) h (y)$ . By separating the variables, the equation follows $\frac{dy}{h (y)} = g (x) dx$ . Then, on direct integration of both sides, the solution of the differential equation is determined.

Step 2: Solution of the Equation

The given equation is

$\frac{dx}{dt} = \frac{t}{x e^{t + 2 x}} \cdot \cdot \cdot \cdot \cdot \cdot (1)$

After separating the variables, equation (1) can be written as

$\begin{matrix} \frac{dx}{dt} = \frac{t}{x e^{t} e^{2 x}} \\ x e^{2 x} dx = t e^{- t} dt \cdot \cdot \cdot \cdot \cdot \cdot (2) \end{matrix}$

Integrate both sides of equation (2). It results,

$\begin{matrix} \int x e^{2 x} dx = \int t e^{- t} dt \\ \frac{1}{2} \int 2 x e^{2 x} dx = (- 1) \int - t e^{- t} dt \end{matrix}$

$\begin{matrix} \frac{1}{2} \int x d (e^{2 x}) = (- 1) \int t d (e^{- t}) \\ \frac{1}{2} [x e^{2 x} - \int e^{2 x} dx] = (- 1) [t e^{- t} - \int e^{- t} dt] [\int u dv = uv - \int v du] \\ \frac{1}{2} [x e^{2 x} - \frac{e^{2 x}}{2}] = (- 1) [t e^{- t} + e^{- t}] + k [k = IntegratingConstant] \\ e^{2 x} (2 x - 1) = - 4 e^{- t} (t + 1) + 4 k \\ e^{2 x} (2 x - 1) + 4 e^{- t} (t + 1) = C [C = 4 k = Constant] \end{matrix}$

Therefore, the solution of the given equation is $e^{2 x} (2 x - 1) + 4 e^{- t} (t + 1) = C$ .

Find Study Materials for

Create Study Materials

Select your language

Fundamentals Of Differential Equations And Boundary Value Problems

Answers without the blur.

Short Answer

In problem 7-16, solve the equation. $\frac{dx}{dt} = \frac{t}{x e^{t + 2 x}}$

Step by Step Solution

Step 1: Concept of Separable Differential Equation

Step 2: Solution of the Equation

Most popular questions for Math Textbooks

Want to see more solutions like these?

Recommended explanations on Math Textbooks

Select your language

Fundamentals Of Differential Equations And Boundary Value Problems

Answers without the blur.

Short Answer

In problem 7-16, solve the equation. dxdt=tx et+2x

Step by Step Solution

Step 1: Concept of Separable Differential Equation

Step 2: Solution of the Equation

Most popular questions for Math Textbooks

Want to see more solutions like these?

Recommended explanations on Math Textbooks

Decision Maths

Calculus

Probability and Statistics

Statistics

Mechanics Maths

Geometry

Pure Maths

In problem 7-16, solve the equation. $\frac{dx}{dt} = \frac{t}{x e^{t + 2 x}}$