A. A one-pound pack of wall putty costs $3. A 10-pound pack costs $24. What is the percent savings If you buy the 10-pound pack?<\/p>\n\n\n\n
a. 20%<\/p>\n\n\n\n
b. 24%<\/p>\n\n\n\n
c. 25%<\/p>\n\n\n\n
d. 60%<\/p>\n\n\n\n
B. Solve the differential equation.<\/p>\n\n\n\n
x2y” + 5xy’ + 4y = 0<\/p>\n\n\n\n
The correct answer and explanation is:<\/mark><\/strong><\/p>\n\n\n\n To calculate the percent savings when buying the 10-pound pack compared to the cost of buying 10 individual 1-pound packs:<\/p>\n\n\n\n Correct answer<\/strong>: We solve the given second-order linear differential equation: This is a Cauchy-Euler equation<\/strong> because the coefficients of (y”), (y’), and (y) are powers of (x).<\/p>\n\n\n\n Assume a solution of the form (y = x^r), where (r) is a constant to be determined. Substitute these into the differential equation: Simplify: Factor (x^r) (non-zero for (x > 0)): Expand and combine terms: Factorize: Thus, (r = -2) is a repeated root<\/strong>.<\/p>\n\n\n\n For a repeated root (r = -2), the general solution is: The given differential equation is a second-order linear Cauchy-Euler equation, characterized by variable coefficients proportional to powers of (x). These equations are efficiently solved using the substitution (y = x^r), which transforms the original equation into an algebraic equation called the characteristic equation<\/strong>.<\/p>\n\n\n\n In this problem, substituting (y = x^r) led to the characteristic equation (r^2 + 4r + 4 = 0), which is quadratic. Solving this yields (r = -2) as a repeated root. When the characteristic equation has repeated roots, the general solution incorporates a logarithmic term, giving (y(x) = C_1 x^{-2} + C_2 x^{-2} \\ln(x)).<\/p>\n\n\n\n This solution represents the superposition of two independent solutions: one proportional to (x^{-2}) and another to (x^{-2} \\ln(x)). These forms arise from the underlying properties of second-order differential equations and their solutions in terms of linearly independent functions.<\/p>\n\n\n\n The method and solution reflect the importance of matching the solution’s form to the equation’s structure. Cauchy-Euler equations frequently model physical systems with scale-invariant properties, such as heat conduction or wave propagation in certain media.<\/p>\n","protected":false},"excerpt":{"rendered":" A. A one-pound pack of wall putty costs $3. A 10-pound pack costs $24. What is the percent savings If you buy the 10-pound pack? a. 20% b. 24% c. 25% d. 60% B. Solve the differential equation. x2y” + 5xy’ + 4y = 0 The correct answer and explanation is: Part A: Percent Savings […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-184213","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/184213","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=184213"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/184213\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=184213"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=184213"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=184213"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}Part A: Percent Savings<\/h3>\n\n\n\n
\n
[
10 \\times 3 = 30 \\, \\text{dollars}.
]<\/li>\n\n\n\n
[
24 \\, \\text{dollars}.
]<\/li>\n\n\n\n
[
30 – 24 = 6 \\, \\text{dollars}.
]<\/li>\n\n\n\n
[
\\left( \\frac{\\text{Savings}}{\\text{Cost of 10 individual packs}} \\right) \\times 100 = \\left( \\frac{6}{30} \\right) \\times 100 = 20\\%.
]<\/li>\n<\/ol>\n\n\n\n
a. 20%<\/strong><\/p>\n\n\n\n
\n\n\n\nPart B: Differential Equation Solution<\/h3>\n\n\n\n
[
x^2 y” + 5x y’ + 4y = 0.
]<\/p>\n\n\n\nStep 1: Identify the type of equation<\/h4>\n\n\n\n
Step 2: Substitute (y = x^r)<\/h4>\n\n\n\n
Calculate derivatives:
[
y’ = r x^{r-1}, \\quad y” = r(r-1)x^{r-2}.
]<\/p>\n\n\n\n
[
x^2 \\cdot r(r-1)x^{r-2} + 5x \\cdot r x^{r-1} + 4x^r = 0.
]<\/p>\n\n\n\n
[
r(r-1)x^r + 5r x^r + 4x^r = 0.
]<\/p>\n\n\n\n
[
r(r-1) + 5r + 4 = 0.
]<\/p>\n\n\n\nStep 3: Solve the characteristic equation<\/h4>\n\n\n\n
[
r^2 – r + 5r + 4 = 0 \\quad \\Rightarrow \\quad r^2 + 4r + 4 = 0.
]<\/p>\n\n\n\n
[
(r + 2)^2 = 0.
]<\/p>\n\n\n\nStep 4: General solution<\/h4>\n\n\n\n
[
y(x) = C_1 x^{-2} + C_2 x^{-2} \\ln(x),
]
where (C_1) and (C_2) are arbitrary constants.<\/p>\n\n\n\n
\n\n\n\nExplanation <\/h3>\n\n\n\n