Chapter 2: Q11E (page 64)

In Problems 9–20, determine whether the equation is exact.
If it is, then solve it.
$(e^{x} \sin y - 3 x^{2}) dx + (e^{x} \cos y + \frac{y^{- \frac{2}{3}}}{3}) dy = 0$

Short Answer

Expert verified

The solution is $e^{x} \sin y - x^{3} + \sqrt[3]{y} = C$ .

Step by step solution

Evaluate whether the equation is exact

Here $(e^{x} \sin y - 3 x^{2}) dx + (e^{x} \cos y + \frac{y^{- \frac{2}{3}}}{3}) dy = 0$

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

$M (x, y) = (e^{x} \sin y - 3 x^{2}) N (x, y) = (e^{x} \cos y + \frac{y^{- \frac{2}{3}}}{3}) \frac{\partial M}{\partial y} = e^{x} \cos y = \frac{\partial N}{\partial x}$

This equation is exact.

Find the value of F (x, y)

Here

$M (x, y) = (e^{x} \sin y - 3 x^{2}) F (x, y) = \int M (x, y) dx + g (y) = \int (e^{x} \sin y - 3 x^{2}) dx + g (y) = e^{x} \sin y - x^{3} + g (y)$

Determine the value of g(y)

$\frac{\partial F}{\partial y} (x, y) = N (x, y) e^{x} \cos y + g' (y) = (e^{x} \cos y + \frac{y^{- \frac{2}{3}}}{3}) g' (y) = \frac{y^{- \frac{2}{3}}}{3} g (y) = \sqrt[3]{y} + C_{1}$

Now $F (x, y) = e^{x} \sin y - x^{3} + \sqrt[3]{y} + C_{1}$

The general solution of the differential equation is $e^{x} \sin y - x^{3} + \sqrt[3]{y} = C$

Hence, the solution is role="math" localid="1664171463132" $e^{x} \sin y - x^{3} + \sqrt[3]{y} = C$

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In Problems 9–20, determine whether the equation is exact.
If it is, then solve it.
$(e^{x} \sin y - 3 x^{2}) dx + (e^{x} \cos y + \frac{y^{- \frac{2}{3}}}{3}) dy = 0$

Short Answer

Step by step solution

Evaluate whether the equation is exact

Find the value of F (x, y)

Determine the value of g(y)

One App. One Place for Learning.

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In Problems 9–20, determine whether the equation is exact.If it is, then solve it.(exsiny-3x2)dx+excosy+y-233dy=0

Short Answer

Step by step solution

Evaluate whether the equation is exact

Find the value of F (x, y)

Determine the value of g(y)

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Theoretical and Mathematical Physics

Decision Maths

Mechanics Maths

Logic and Functions

Geometry

Study anywhere. Anytime. Across all devices.

In Problems 9–20, determine whether the equation is exact.
If it is, then solve it.
$(e^{x} \sin y - 3 x^{2}) dx + (e^{x} \cos y + \frac{y^{- \frac{2}{3}}}{3}) dy = 0$