True-False Review
For Questions (a)-(i), decide if the given statement is true or false, and give a brief justification for your answer. If true, you can quote a relevant definition or theorem in fact from the text. If false, provide an example, illustration, or brief explanation of why the statement is false.
(a) The function
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Differential Equations and Linear Algebra (4th Edition)
- Hollings Functional Response Curve The total number P of prey taken by a predator depends on the availability of prey. C.S. Holling proposed a function of the form P=cn(1+dn) to model the number of prey taken in certain situations. Here n is the density of prey available, and c and d are constants that depend on the organisms involved as well as on other environmental features. Holling took data gathered earlier by T. Burnett on the number of sawfly cocoons found by a small wasp parasite at given host density. In one such experiment conducted, Holling found the relationship p=21.96n1+2.41n, Where P is the number of cocoons parasitized and n is the density of cocoons available measured as number per square inch. a Draw a graph of p versus n. Include values of n up to 2 cocoons per square inch. b What density of cocoons will ensure that the wasp will find and parasitize 6 of them? c There is a limit to the number of cocoons that the wasp is able to parasitize no matter how readily available the prey may be. What is this upper limit?arrow_forwardDoing practice problems from the textbook. What are the critical points of f(x,y) = (e^x)(sin(y)-1)arrow_forwardHomogeneous DE M(x,y)dx + N(x,y)dy = 0 is Homogeneous if both M and N are homogeneous functions of the same order. A function f(x,y) is called homogeneous of degree n if f(ʎx, ʎy) = ʎnf(x,y) Solution Steps: Objective: To reduce into a Variable Separable Form a. Replace y by ux or x by vy if y = ux, dy = udx + xdu . Use y=ux if N is simpler in form. if x = vy, dx = vdy + ydv . Use x=y if M is simpler in form b. Simplify the resulting equation c. Separate the variables d. Integrate both sides of the equation to get the General Solution e. Substitute u = y/x or v = x/y to have the GS in terms of x and y have C which is arbitrary constant Solve the DE. (Homogeneous DE) 1. xdx + sin2 (y/x) [ydx - xdy = 0]arrow_forward
- Homogeneous DE M(x,y)dx + N(x,y)dy = 0 is Homogeneous if both M and N are homogeneous functions of the same order. A function f(x,y) is called homogeneous of degree n if f(ʎx, ʎy) = ʎnf(x,y) Solution Steps: Objective: To reduce into a Variable Separable Form a. Replace y by ux or x by vy if y = ux, dy = udx + xdu . Use y=ux if N is simpler in form. if x = vy, dx = vdy + ydv . Use x=y if M is simpler in form b. Simplify the resulting equation c. Separate the variables d. Integrate both sides of the equation to get the General Solution e. Substitute u = y/x or v = x/y to have the GS in terms of x and y have C which is arbitrary constant Solve the DE. (Homogeneous DE) 1. x2y' = 4x2 + 7xy + 2y2arrow_forwardHomogeneous DE M(x,y)dx + N(x,y)dy = 0 is Homogeneous if both M and N are homogeneous functions of the same order. A function f(x,y) is called homogeneous of degree n if f(ʎx, ʎy) = ʎnf(x,y) Solution Steps: Objective: To reduce into a Variable Separable Form a. Replace y by ux or x by vy if y = ux, dy = udx + xdu . Use y=ux if N is simpler in form. if x = vy, dx = vdy + ydv . Use x=y if M is simpler in form b. Simplify the resulting equation c. Separate the variables d. Integrate both sides of the equation to get the General Solution e. Substitute u = y/x or v = x/y to have the GS in terms of x and y have C which is arbitrary constant Solve the DE. (Homogeneous DE) 1. [y - (x2+y2)1/2]dx - xdy = 0arrow_forwardHomogeneous DE M(x,y)dx + N(x,y)dy = 0 is Homogeneous if both M and N are homogeneous functions of the same order. A function f(x,y) is called homogeneous of degree n if f(ʎx, ʎy) = ʎnf(x,y) Solution Steps: Objective: To reduce into a Variable Separable Form a. Replace y by ux or x by vy if y = ux, dy = udx + xdu . Use y=ux if N is simpler in form. if x = vy, dx = vdy + ydv . Use x=y if M is simpler in form b. Simplify the resulting equation c. Separate the variables d. Integrate both sides of the equation to get the General Solution e. Substitute u = y/x or v = x/y to have the GS in terms of x and y have C which is arbitrary constant Solve the DE. (Homogeneous DE) 1. (x3 + y3) dx + 3xy2dy = 0arrow_forward
- Homogeneous DE M(x,y)dx + N(x,y)dy = 0 is Homogeneous if both M and N are homogeneous functions of the same order. A function f(x,y) is called homogeneous of degree n if f(ʎx, ʎy) = ʎnf(x,y) Solution Steps: Objective: To reduce into a Variable Separable Form a. Replace y by ux or x by vy if y = ux, dy = udx + xdu . Use y=ux if N is simpler in form. if x = vy, dx = vdy + ydv . Use x=y if M is simpler in form b. Simplify the resulting equation c. Separate the variables d. Integrate both sides of the equation to get the General Solution e. Substitute u = y/x or v = x/y to have the GS in terms of x and y have C which is arbitrary constant Solve the DE. (Homogeneous DE) 1. (y2-x2)dx + xydy = 0arrow_forwardHomogeneous DE M(x,y)dx + N(x,y)dy = 0 is Homogeneous if both M and N are homogeneous functions of the same order. A function f(x,y) is called homogeneous of degree n if f(ʎx, ʎy) = ʎnf(x,y) Solution Steps: Objective: To reduce into a Variable Separable Form a. Replace y by ux or x by vy if y = ux, dy = udx + xdu . Use y=ux if N is simpler in form. if x = vy, dx = vdy + ydv . Use x=y if M is simpler in form b. Simplify the resulting equation c. Separate the variables d. Integrate both sides of the equation to get the General Solution e. Substitute u = y/x or v = x/y to have the GS in terms of x and y have C which is arbitrary constant Solve the DE. (Homogeneous DE) 1. [xcos2(y/x) - y]dx + xdy = 0arrow_forwardHomogeneous DE M(x,y)dx + N(x,y)dy = 0 is Homogeneous if both M and N are homogeneous functions of the same order. A function f(x,y) is called homogeneous of degree n if f(ʎx, ʎy) = ʎnf(x,y) Solution Steps: Objective: To reduce into a Variable Separable Form a. Replace y by ux or x by vy if y = ux, dy = udx + xdu . Use y=ux if N is simpler in form. if x = vy, dx = vdy + ydv . Use x=y if M is simpler in form b. Simplify the resulting equation c. Separate the variables d. Integrate both sides of the equation to get the General Solution e. Substitute u = y/x or v = x/y to have the GS in terms of x and y have C which is arbitrary constant Solve the DE. (Homogeneous DE) 1. ydx = [x + (y2 - x2)1/2]dyarrow_forward
- Homogeneous DE M(x,y)dx + N(x,y)dy = 0 is Homogeneous if both M and N are homogeneous functions of the same order. A function f(x,y) is called homogeneous of degree n if f(ʎx, ʎy) = ʎnf(x,y) Solution Steps: Objective: To reduce into a Variable Separable Form a. Replace y by ux or x by vy if y = ux, dy = udx + xdu . Use y=ux if N is simpler in form. if x = vy, dx = vdy + ydv . Use x=y if M is simpler in form b. Simplify the resulting equation c. Separate the variables d. Integrate both sides of the equation to get the General Solution e. Substitute u = y/x or v = x/y to have the GS in terms of x and y have C which is arbitrary constant Solve the DE. (Homogeneous DE) 1. xydx - (x + 2y)2dy - 0arrow_forwardContinuity At what points of ℝ2 are the following functions continuous? ƒ(x, y) = x2 + 2xy - y3arrow_forwarddiiferential of a function: f(x,y)=ye^x at (0,-2)arrow_forward
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