Chapter 1: Q13E (page 1)

In Problems 9–20, determine whether the equation is exact.
If it is, then solve it.
$e^{t} (y - t) dt + ({1 +e}^{t}) dy = 0$

Short Answer

Expert verified

The solution is $y = \frac{(t - 1) e^{t} + C}{1 + e^{t}}$ .

Step by step solution

Evaluate whether the equation is exact

Here $e^{t} (y - t) dt + ({1 +e}^{t}) dy = 0$

The condition for exact is $\frac{\partial M}{\partial y} = \frac{\partial N}{\partial t}$ .

$M (t, y) = e^{t} (y - t) N (t, y) = (1 + e^{t})$

$\frac{\partial M}{\partial y} = e^{t} = \frac{\partial N}{\partial t}$

This equation is exact.

Find the value of F (x, t)

Here

$M (t, y) = e^{t} (y - t) F (t, y) = \int M (t, y) dt + g (y) = \int e^{t} (y - t) dt + g (y) = e^{t} y - e^{t} t + e^{t} + g (y)$

Determine the value of g(y)

$\frac{\partial F}{\partial y} (t, y) = N (t, y) e^{t} + g' (y) = 1 + e^{t} g' (y) = 1 g (y) = y + C_{1}$

Now ${F(t,y) =e}^{t} {y -e}^{t} {t +e}^{t} {+ y +C}_{1}$

The general solution of the differential equation is $y = \frac{(t - 1) e^{t} + C}{1 + e^{t}}$

Hence the solution isrole="math" localid="1664172391787" $y = \frac{(t - 1) e^{t} + C}{1 + e^{t}}$

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In Problems 9–20, determine whether the equation is exact.
If it is, then solve it.
$e^{t} (y - t) dt + ({1 +e}^{t}) dy = 0$

Short Answer

Step by step solution

Evaluate whether the equation is exact

Find the value of F (x, t)

Determine the value of g(y)

One App. One Place for Learning.

Most popular questions from this chapter

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In Problems 9–20, determine whether the equation is exact.If it is, then solve it.et(y - t)dt +(1 +et)dy = 0

Short Answer

Step by step solution

Evaluate whether the equation is exact

Find the value of F (x, t)

Determine the value of g(y)

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Pure Maths

Geometry

Applied Mathematics

Mechanics Maths

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

In Problems 9–20, determine whether the equation is exact.
If it is, then solve it.
$e^{t} (y - t) dt + ({1 +e}^{t}) dy = 0$