Explanation Given information: ∫ − 1 1 ( ∫ − 2 0 x e x y d y ) d x Formula used: Let f ( x , y ) be a function of two variables, let g ( x ) and h ( x ) be two functions of x alone, and let a and b be numbers. Then, an iterated integral is of the form ∫ a b ( ∫ g ( x ) h ( x ) f ( x , y ) d y ) d x . The integral ∫ g ( x ) h ( x ) f ( x , y ) d y be considering f ( x , y ) as a function of y alone. If F ( x , y ) is an antiderivative of f ( x , y ) with respect to y then the above integral is evaluated as F ( x , h ( x ) ) − F ( x , g ( x ) ) . Evaluate the antiderivative between the limits y = g ( x ) and y = h ( x ) . This gives the function in x alone then integrate that integral from x = a to x = b . Calculation: Here g ( x ) and h ( x ) are constant functions. The limits of inner integral are g ( x ) = − 2 and h ( x ) = 0 . Evaluate the inner integral first. The variable in this inner integral is y , so treat x as a constant. ∫ − 2 0 x e x y d y = x ∫ − 2 0 e x y d y = x ( e x y x | − 2 0 ) Simplify, = x x (

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Chapter 7.6, Problem 3E

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To calculate: The iterated integral ∫−11(∫−20xexydy) dx.

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Chapter 7.6, Problem 2E

Chapter 7.6, Problem 4E

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1. Abel's Theorem. The goal in this problem is to prove Abel's theorem by following a series of steps (each step must be justified). Theorem 0.1 (Abel's Theorem). If y1 and y2 are solutions of the differential equation y" + p(t) y′ + q(t) y = 0, where p and q are continuous on an open interval, then the Wronskian is given by W (¥1, v2)(t) = c exp(− [p(t) dt), where C is a constant that does not depend on t. Moreover, either W (y1, y2)(t) = 0 for every t in I or W (y1, y2)(t) = 0 for every t in I. 1. (a) From the two equations (which follow from the hypotheses), show that y" + p(t) y₁ + q(t) y₁ = 0 and y½ + p(t) y2 + q(t) y2 = 0, 2. (b) Observe that Hence, conclude that (YY2 - Y1 y2) + P(t) (y₁ Y2 - Y1 Y2) = 0. W'(y1, y2)(t) = yY2 - Y1 y2- W' + p(t) W = 0. 3. (c) Use the result from the previous step to complete the proof of the theorem.

2. Observations on the Wronskian. Suppose the functions y₁ and y2 are solutions to the differential equation p(x)y" + q(x)y' + r(x) y = 0 on an open interval I. 1. (a) Prove that if y₁ and y2 both vanish at the same point in I, then y₁ and y2 cannot form a fundamental set of solutions. 2. (b) Prove that if y₁ and y2 both attain a maximum or minimum at the same point in I, then y₁ and Y2 cannot form a fundamental set of solutions. 3. (c) show that the functions & and t² are linearly independent on the interval (−1, 1). Verify that both are solutions to the differential equation t² y″ – 2ty' + 2y = 0. Then justify why this does not contradict Abel's theorem. 4. (d) What can you conclude about the possibility that t and t² are solutions to the differential equation y" + q(x) y′ + r(x)y = 0?

Question 4 Find an equation of (a) The plane through the point (2, 0, 1) and perpendicular to the line x = y=2-t, z=3+4t. 3t, (b) The plane through the point (3, −2, 8) and parallel to the plane z = x+y. (c) The plane that contains the line x = 1+t, y = 2 − t, z = 4 - 3t and is parallel to the plane 5x + 2y + z = 1. (d) The plane that passes through the point (1,2,3) and contains the line x = 3t, y = 1+t, and z = 2-t. (e) The plane that contains the lines L₁: x = 1 + t, y = 1 − t, z = 2t and L2 : x = 2 − s, y = s, z = 2.

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Ch. 7.1 - Let f(x,y,z)=x2+y(xz)4. Compute f(3,5,2).Ch. 7.1 - Prob. 2CYU Ch. 7.1 - Let f(x,y)=x23xyy2. Compute f(5,0),f(5,2) and...Ch. 7.1 - Prob. 2E Ch. 7.1 - Let g(x,y,z)=x/(yz). Compute g(1,1),g(0,1) and...Ch. 7.1 - Let f(x,y,z)=x2ey2+z2. Compute f(1,1,1) and...Ch. 7.1 - Let f(x,y)=xy. Show that f(2+h,3)f(2,3)=3h.Ch. 7.1 - Let f(x,y)=xy. Show that f(2,3+k)f(2,3)=2k.Ch. 7.1 - Cost Find a formula C(x,y,z) that gives the cost...Ch. 7.1 - Cost Find a formula C(x,y,z) that gives the cost...

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The iterated integral ∫ − 1 1 ( ∫ − 2 0 x e x y d y ) d x .

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To calculate: The iterated integral ∫−11(∫−20xexydy) dx.

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