Chapter 2: Q27E (page 64)

For each of the following equations, find the most general function
so that the equation is exact.
$(a) M (x, y) dx + (\sec^{2} y - \frac{x}{y}) dy = 0 (b) M (x, y) dx + (\sin x \cos y - xy - e^{- y}) dy = 0$

Short Answer

Expert verified

$(a) M (x, y) = - \ln |y| + f (x) (b) M (x, y) = \cos x \sin y - \frac{y^{2}}{2} + f (x)$

Step by step solution

Step 1(a): Find the value of∂M∂y if M(x,y)dx+sec2y-xydy=0

Here $M (x, y) dx + (\sec^{2} y - \frac{x}{y}) dy = 0$

Since the equation is exact.so, the condition for exact is $\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$ .

$\frac{\partial M}{\partial y} = - \frac{1}{y} = \frac{\partial N}{\partial x} \frac{\partial M}{\partial y} = - \frac{1}{y} . . . . . . (1)$

Integrating the equation (1)

$M (x, y) = - \ln |y| + f (x)$

Therefore, the value of M is $M (x, y) = - \ln | y | + f (x)$ .

Step 2(b): Evaluate the value of∂M∂y if M(x,y)dx+(sin x cos y-xy-e-y)dy=0

Here $M (x, y) dx + (\sin x \cos y - xy - e^{- y}) dy = 0$

Since the equation is exact. So, the condition for exact is $\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$ .

$\frac{\partial M}{\partial y} = \cos x \cos y - y = \frac{\partial N}{\partial x} \frac{\partial M}{\partial y} = \cos x \cos y - y . . . . . . (2)$

Integrating the equation (2)

$M (x, y) = \cos x \sin y - \frac{y^{2}}{2} + f (x)$

Therefore, the required result is $M (x, y) = \cos x \sin y - \frac{y^{2}}{2} + f (x)$ .

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For each of the following equations, find the most general function
so that the equation is exact.
$(a) M (x, y) dx + (\sec^{2} y - \frac{x}{y}) dy = 0 (b) M (x, y) dx + (\sin x \cos y - xy - e^{- y}) dy = 0$

Short Answer

Step by step solution

Step 1(a): Find the value of∂M∂y if M(x,y)dx+sec2y-xydy=0

Step 2(b): Evaluate the value of∂M∂y if M(x,y)dx+(sin x cos y-xy-e-y)dy=0

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For each of the following equations, find the most general functionso that the equation is exact.(a)M(x,y)dx+(sec2y-xy)dy=0(b)M(x,y)dx+(sinxcosy-xy-e-y)dy=0

Short Answer

Step by step solution

Step 1(a): Find the value of∂M∂y if M(x,y)dx+sec2y-xydy=0

Step 2(b): Evaluate the value of∂M∂y if M(x,y)dx+(sin x cos y-xy-e-y)dy=0

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Pure Maths

Probability and Statistics

Logic and Functions

Decision Maths

Mechanics Maths

Study anywhere. Anytime. Across all devices.

For each of the following equations, find the most general function
so that the equation is exact.
$(a) M (x, y) dx + (\sec^{2} y - \frac{x}{y}) dy = 0 (b) M (x, y) dx + (\sin x \cos y - xy - e^{- y}) dy = 0$