Chapter 2: Q20E (page 64)

In Problems 9–20, determine whether the equation is exact.
If it is, then solve it.
$(\frac{2}{\sqrt{1 - x^{2}}} + ycos (xy)) dx + (xcos (xy) - y^{- \frac{1}{3}}) dy = 0$

Short Answer

Expert verified

The solution is $2 \sin^{- 1} x + \sin xy - \frac{3}{2} y^{\frac{2}{3}} = C$ .

Step by step solution

Evaluate the equation is exact

Here $(\frac{2}{\sqrt{1 - x^{2}}} + y \cos (xy)) dx + (x \cos (xy) - y^{- \frac{1}{3}}) dy = 0$

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

$M (x, y) = (\frac{2}{\sqrt{1 - x^{2}}} + y \cos (xy)) N (x, y) = (x \cos (xy) - y^{- \frac{1}{3}})$

$\frac{\partial M}{\partial y} = (\cos (xy) - x y \sin (xy)) = \frac{\partial N}{\partial x}$

This equation is exact.

Find the value of F(x, y)

Here

M (x, y) = (\frac{2}{\sqrt{1 - x^{2}}} + y \cos (xy)) F (x, y) = \int M (x, y) dx + g (y) = \int (\frac{2}{\sqrt{1 - x^{2}}} + y \cos (xy)) dx + g (y) = 2 \sin^{- 1} x + \sin (xy) + g (y)

Determine the value of g(y)

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

Now $F (x, y) = 2 si n^{- 1} x + \sin xy - \frac{3}{2} y^{\frac{2}{3}} + C_{1}$

The solution of the differential equation is $2 \sin^{- 1} x + \sin xy - \frac{3}{2} y^{\frac{2}{3}} = C$ .

Hence the solution isrole="math" localid="1664178541241" $2 \sin^{- 1} x + \sin xy - \frac{3}{2} y^{\frac{2}{3}} = C$

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In Problems 9–20, determine whether the equation is exact.
If it is, then solve it.
$(\frac{2}{\sqrt{1 - x^{2}}} + ycos (xy)) dx + (xcos (xy) - y^{- \frac{1}{3}}) dy = 0$

Short Answer

Step by step solution

Evaluate 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

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In Problems 9–20, determine whether the equation is exact.If it is, then solve it.21-x2+ycos(xy)dx+xcos(xy)-y-13dy=0

Short Answer

Step by step solution

Evaluate 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

Calculus

Statistics

Pure Maths

Mechanics Maths

Study anywhere. Anytime. Across all devices.

In Problems 9–20, determine whether the equation is exact.
If it is, then solve it.
$(\frac{2}{\sqrt{1 - x^{2}}} + ycos (xy)) dx + (xcos (xy) - y^{- \frac{1}{3}}) dy = 0$