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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+(sec2y-xy)dy=0(b)M(x,y)dx+(sinxcosy-xy-e-y)dy=0

Short Answer

Expert verified

(a)M(x,y)=-lny+f(x)(b)M(x,y)=cosxsiny-y22+f(x)

Step by step solution

01

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

HereM(x,y)dx+sec2y-xydy=0

Since the equation is exact.so, the condition for exact is∂M∂y=∂N∂x .

∂M∂y=-1y=∂N∂x∂M∂y=-1y  ......(1)

Integrating the equation (1)

M (x,y)=-ln y+f (x)

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

02

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

HereM (x,y) dx+(sin x cos y-xy-e-y) dy=0

Since the equation is exact. So, the condition for exact is ∂M∂y=∂N∂x.

∂M∂y=cos x cos y-y=∂N∂x∂M∂y=cosxcosy-y ......(2)

Integrating the equation (2)

M (x,y)=cos x sin y-y22+f (x)

Therefore, the required result isM (x,y)=cosx sin y-y22+f (x).

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Most popular questions from this chapter

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