Chapter 2: Q 33RP (page 79)

Question: In Problems 31-40, solve the initial value problem.
$(t + x + 3) dt + dx = 0, x (0) = 1$

Short Answer

Expert verified

The solution of the given equation is $x = - t - 2 + 3 e^{- t}$ .

Step by step solution

Given information and simplification

Given that, $(t + x + 3) dt + dx = 0, x (0) = 1 \cdot \cdot \cdot \cdot \cdot \cdot (1)$

Evaluate the equation (1).

$\begin{matrix} (t + x + 3) dt + dx = 0 \\ dx = - (t + x + 3) dt \\ \frac{dx}{dt} = - (t + x + 3) \end{matrix}$

Let us assume $p = t + x + 3$ . Differentiate with respect to t.

$\frac{dp}{dt} = 1 + \frac{dx}{dt}$

Substitute the values in equation (1)

$\begin{matrix} \frac{dp}{dt} - 1 = - p \\ \frac{dp}{dt} = - p + 1 \\ \frac{1}{1 - p} dp = dt \end{matrix}$

Now integrate the equation (2) on both sides.

$\begin{matrix} \int \frac{1}{1 - p} dp = \int dt \\ \frac{\ln |1 - p|}{- 1} = t + C_{1} \\ \ln |1 - p| = - t + C_{1} \\ 1 - p = {Ce}^{- t} \end{matrix}$

Find the initial value

Substitute the value of p.

\begin{matrix} 1 - (t + x + 3) = {Ce}^{- t} \\ t + x + 2 = {Ce}^{- t} \end{matrix}

So, the solution is found.

Given that, x ( 0 ) = 1.

Then, x = 1 and t = 0.

Substitute the value in equation (4) to get the value of C.

$\begin{matrix} t + x + 2 = {Ce}^{- t} \\ 0 + 1 + 2 = {Ce}^{0} \\ 3 = C \end{matrix}$

Substitute the value of C in equation (2).

$\begin{matrix} t + x + 2 = 3 e^{- t} \\ x = - t - 2 + 3 e^{- t} \end{matrix}$

So, the solution is $x = - t - 2 + 3 e^{- t}$

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Question: In Problems 31-40, solve the initial value problem.
$(t + x + 3) dt + dx = 0, x (0) = 1$

Short Answer

Step by step solution

Given information and simplification

Find the initial value

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Question: In Problems 31-40, solve the initial value problem.t+x+3dt+dx=0, x0=1

Short Answer

Step by step solution

Given information and simplification

Find the initial value

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Calculus

Theoretical and Mathematical Physics

Decision Maths

Applied Mathematics

Statistics

Study anywhere. Anytime. Across all devices.

Question: In Problems 31-40, solve the initial value problem.
$(t + x + 3) dt + dx = 0, x (0) = 1$