{"id":32179,"date":"2024-12-11T06:52:55","date_gmt":"2024-12-11T01:22:55","guid":{"rendered":"https:\/\/samacheerkalvi.guide\/?p=32179"},"modified":"2024-12-12T10:19:11","modified_gmt":"2024-12-12T04:49:11","slug":"samacheer-kalvi-12th-business-maths-guide-chapter-4-ex-4-6","status":"publish","type":"post","link":"https:\/\/samacheerkalvi.guide\/samacheer-kalvi-12th-business-maths-guide-chapter-4-ex-4-6\/","title":{"rendered":"Samacheer Kalvi 12th Business Maths Guide Chapter 4 Differential Equations Ex 4.6"},"content":{"rendered":"

Tamilnadu State Board New Syllabus\u00a0Samacheer Kalvi 12th Business Maths Guide<\/a> Pdf Chapter 4 Differential Equations Ex 4.6 Text Book Back Questions and Answers, Notes.<\/p>\n

Tamilnadu Samacheer Kalvi 12th Business Maths Solutions Chapter 4 Differential Equations Ex 4.6<\/h2>\n

Choose the most suitable answer from the given four alternatives:<\/span><\/p>\n

Question 1.
\nThe degree of the differential equation
\n\\(\\frac { d^2y }{dx^4}\\) – (\\(\\frac { d^2y }{dx^2}\\)) + \\(\\frac { dy }{dx}\\) = 3
\n(a) 1
\n(b) 2
\n(c) 3
\n(d) 4
\nSolution:
\n(a) 1
\nHint:
\nSince the power of \\(\\frac{d^{4} y}{d x^{4}}\\) is 1<\/p>\n

\"Samacheer<\/p>\n

Question 2.
\nThe order and degree of the differential equation \\(\\sqrt{\\frac { d^2y }{dx^2}}\\) = \\(\\sqrt{\\frac { dy }{dx}+5}\\) are respectively.
\n(a) 2 and 2
\n(b) 3 and 2
\n(c) 2 and 1
\n(d) 2 and 3
\nSolution:
\n(a) 1
\nHint:
\nSquaring on both sides
\n\\(\\frac { d^2y }{dx^2}\\) = \\(\\frac { dy }{dx}\\) + 5
\nHighest order derivative is \\(\\frac { d^2y }{dx^2}\\)
\n\u2234 order = 2
\nPower of the highest order derivative \\(\\frac { d^2y }{dx^2}\\) = 1
\n\u2234 degree = 1<\/p>\n

Question 3.
\nThe order and degree of the differential equation
\n(\\(\\frac { d^2y }{dx^2}\\))3\/2<\/sup> – \\(\\sqrt{(\\frac { dy }{dx})}\\) – 4 = 0
\n(a) 2 and 6
\n(b) 3 and 6
\n(c) 1 and 4
\n(d) 2 and 4
\nSolution:
\n(a) 2 and 6
\nHint:
\n\"Samacheer
\nHighest order derivative is \\(\\frac { d^2y }{dx^2}\\)
\n\u2234 Order = 2
\nPower of the highest order derivative \\(\\frac { d^2y }{dx^2}\\) is
\n\u2234 degree = 6<\/p>\n

\"Samacheer<\/p>\n

Question 4.
\nThe differential equation (\\(\\frac { dx }{dy}\\))\u00b3 + 2y1\/2<\/sup> = x
\n(a) of order 2 and degree 1
\n(b) of order 1 and degree 3
\n(c) of order 1 and degree 6
\n(d) of order 1 and degree 2
\nSolution:
\n(b) of order 1 and degree 3
\nHint:
\n\"Samacheer
\nHighest order derivative is \\(\\frac { dy }{dx}\\)
\n\u2234 order = 1
\nPower of the highest order derivative \\(\\frac { dy }{dx}\\) is 3
\n\u2234 degree = 3<\/p>\n

Question 5.
\nThe differential equation formed by eliminating a and b from y = aex<\/sup> + be-x<\/sup>
\n(a) \\(\\frac { d^2y }{dx^2}\\) – y = 0
\n(b) \\(\\frac { d^2y }{dx^2}\\) – \\(\\frac { dy }{dx}\\)y = 0
\n(c) \\(\\frac { d^2y }{dx^2}\\) = 0
\n(d) \\(\\frac { d^2y }{dx^2}\\) – x = 0
\nSolution:
\n(a) \\(\\frac { d^2y }{dx^2}\\) – y = 0
\nHint:
\n\"Samacheer<\/p>\n

\"Samacheer<\/p>\n

Question 6.
\nIf y = ex + c – c\u00b3 then its differential equation is
\n(a) y = x\\(\\frac { dy }{dx}\\) + \\(\\frac { dy }{dx}\\) – (\\(\\frac { dy }{dx}\\))\u00b3
\n(b) y + (\\(\\frac { dy }{dx}\\))\u00b3 = x \\(\\frac { dy }{dx}\\) – \\(\\frac { dy }{dx}\\)
\n(c) \\(\\frac { dy }{dx}\\) + (\\(\\frac { dy }{dx}\\))\u00b3 – x\\(\\frac { dy }{dx}\\)
\n(d) \\(\\frac { d^3y }{dx^3}\\) = 0
\nSolution:
\n(a) y = x\\(\\frac { dy }{dx}\\) + \\(\\frac { dy }{dx}\\) – (\\(\\frac { dy }{dx}\\))\u00b3
\nHint:
\ny = cx + c – c\u00b3 ……… (1)
\n\\(\\frac { dy }{dx}\\) = c
\n(1) \u21d2 y = x\\(\\frac { dy }{dx}\\) + \\(\\frac { dy }{dx}\\) – (\\(\\frac { dy }{dx}\\))\u00b3<\/p>\n

Question 7.
\nThe integrating factor of the differential equation \\(\\frac { dy }{dx}\\) + Px = Q is
\n(a) e\u222bpdx<\/sup>
\n(b) e\u222b<\/sup>Pdx
\n(c) e\u222b<\/sup>Pdy
\n(d) e\u222bpdy<\/sup>
\nSolution:
\n(d) e\u222bpdy<\/sup><\/p>\n

\"Samacheer<\/p>\n

Question 8.
\nThe complementary function of (D\u00b2 + 4) y = e2x<\/sup> is
\n(a) (Ax+B)e2x<\/sup>
\n(b) (Ax+B)e-2x<\/sup>
\n(c) A cos2x + B sin2x
\n(d) Ae-2x<\/sup> + Be2x<\/sup>
\nSolution:
\n(c) A cos2x + B sin2x
\nHint:
\nA.E = m2<\/sup> + 4 = 0 \u21d2 m = \u00b12i
\nC.F = e0x<\/sup> (A cos 2x + B sin 2x)<\/p>\n

Question 9.
\nThe differential equation of y = mx + c is (m and c are arbitrary constants)
\n(a) \\(\\frac { d^2y }{dx^2}\\) = 0
\n(b) y = x\\(\\frac { dy }{dx}\\) + o
\n(c) xdy + ydx = 0
\n(c) ydx – xdy = 0
\nSolution:
\n(a) \\(\\frac { d^2y }{dx^2}\\) = 0
\nHint:
\ny = mx + c
\n\\(\\frac { dy }{dx}\\) = m \u21d2 \\(\\frac { d^2y }{dx^2}\\) = 0<\/p>\n

\"Samacheer<\/p>\n

Question 10.
\nThe particular intergral of the differential equation \\(\\frac { d^2y }{dx^2}\\) – 8\\(\\frac { dy }{dx}\\) + 16y = 2e4x<\/sup>
\n(a) \\(\\frac { x^2e^{4x} }{2!}\\)
\n(b) y = x\\(\\frac { e^{4x} }{2!}\\)
\n(c) x\u00b2e4x<\/sup>
\n(d) xe4x<\/sup>
\nSolution:
\n(c) x\u00b2e4x<\/sup>
\nHint:
\n\"Samacheer<\/p>\n

Question 11.
\nSolution of \\(\\frac { dx }{dy}\\) + Px = 0
\n(a) x = cepy<\/sup>
\n(b) x = ce-py<\/sup>
\n(c) x = py + c
\n(d) x = cy
\nSolution:
\n(b) x = ce-py<\/sup><\/p>\n

\"Samacheer<\/p>\n

Question 12.
\nIf sec2x<\/sup> x isa na intergranting factor of the differential equation \\(\\frac { dx }{dy}\\) + Px = Q then P =
\n(a) 2 tan x
\n(b) sec x
\n(c) cos 2 x
\n(d) tan 2 x
\nSolution:
\n(a) 2 tan x
\nHint:
\nI.F = sec\u00b2 x
\ne\u222bpdx<\/sup> = sec\u00b2x
\n\u222bpdx = log sec\u00b2 x
\n\u222bpdx = 2 log sec x
\n\u222bpdx = 2\u222btan x dx \u21d2 p = 2 tan x<\/p>\n

Question 13.
\nThe integrating factor of the differential equation is x \\(\\frac { dy }{dx}\\) – y = x\u00b2
\n(a) \\(\\frac { -1 }{x}\\)
\n(b) \\(\\frac { 1 }{x}\\)
\n(c) log x
\n(c) x
\nSolution:
\n(b) \\(\\frac { 1 }{x}\\)
\nHint:
\n\"Samacheer<\/p>\n

\"Samacheer<\/p>\n

Question 14.
\nThe solution of the differential equation where P and Q are the function of x is
\n(a) y = \u222bQ e\u222bpdx<\/sup> dx + c
\n(b) y = \u222bQ e-\u222bpdx<\/sup> dx + c
\n(c) ye\u222bpdx<\/sup> = \u222bQ e\u222bpdx<\/sup> dx + c
\n(c) ye\u222bpdx<\/sup> = \u222bQ e-\u222bpdx<\/sup> dx + c
\nSolution:
\n(c) ye\u222bpdx<\/sup> = \u222bQ e\u222bpdx<\/sup> dx + c<\/p>\n

Question 15.
\nThe differential equation formed by eliminating A and B from y = e-2x<\/sup> (A cos x + B sin x) is
\n(a) y2<\/sub> – 4y1<\/sub> + 5 = 0
\n(b) y2<\/sub> + 4y – 5 = 0
\n(c) y2<\/sub> – 4y1<\/sub> + 5 = 0
\n(d) y2<\/sub> + 4y1<\/sub> – 5 = 0
\nSolution:
\n(d) y2<\/sub> + 4y1<\/sub> – 5 = 0
\nHint:
\ny = e-2x<\/sup> (A cosx + B sinx)
\ny e2x<\/sup> = A cosx + B sinx ………. (1)
\ny(e2x<\/sup>) (2) + e2x<\/sup> \\(\\frac { dy }{dx}\\) = A(-sin x) + B cos x ………. (2)
\nDifferentiating w.r.to x
\n\"Samacheer<\/p>\n

\"Samacheer<\/p>\n

Question 16.
\nThe particular integral of the differential equation f (D) y = eax<\/sup> where f(D) = (D – a)\u00b2
\n(a) \\(\\frac { x^2 }{2}\\) eax<\/sup>
\n(b) xeax<\/sup>
\n(c) \\(\\frac { x }{2}\\) eax<\/sup>
\n(d) x\u00b2 eax<\/sup>
\nSolution:
\n(a) \\(\\frac { x^2 }{2}\\) eax<\/sup><\/p>\n

Question 17.
\nThe differential equation of x\u00b2 + y\u00b2 = a\u00b2
\n(a) xdy + ydx = 0
\n(b) ydx – xdy = 0
\n(c) xdx – ydx = 0
\n(d) xdx + ydy = 0
\nSolution:
\n(d) xdx + ydy = 0
\nHint:
\nx2<\/sup> + y2<\/sup> = a2<\/sup>
\n\u21d2 2x + 2y \\(\\frac{d y}{d x}\\) = 0
\n\u21d2 x dx + y dy = 0<\/p>\n

\"Samacheer<\/p>\n

Question 18.
\nThe complementary function of \\(\\frac { d^y }{dx^2}\\) – \\(\\frac { dy }{dx}\\) = 0 is
\n(a) A + Bex<\/sup>
\n(b) (A + B)ex<\/sup>
\n(c) (Ax + B)ex<\/sup>
\n(d) Aex<\/sup> + B
\nSolution:
\n(a) A + Bex<\/sup>
\nHint:
\nA.E is m2<\/sup> – m = 0
\n\u21d2 m(m – 1) = 0
\n\u21d2 m = 0, 1
\nCF is Ae0x<\/sup> + Bex<\/sup> = A + Bex<\/sup><\/p>\n

Question 19.
\nThe P.I of (3D\u00b2 + D – 14) y = 13e2x<\/sup> is
\n(a) \\(\\frac { 1 }{2}\\) ex<\/sup>
\n(b) xe2x<\/sup>
\n(c) \\(\\frac { x^2 }{2}\\) e2x<\/sup>
\n(d) Aex<\/sup> + B
\nSolution:
\n(b) xe2x<\/sup>
\nHint:
\n(3D\u00b2 + D – 14) y = 13e2x<\/sup>
\n\"Samacheer<\/p>\n

\"Samacheer<\/p>\n

Question 20.
\nThe general solution of the differential equation \\(\\frac { dy }{dx}\\) = cos x is
\n(a) y = sinx + 1
\n(b) y = sinx – 2
\n(c) y = cosx + C, C is an arbitary constant
\n(d) y = sinx + C, C is an arbitary constant
\nSolution:
\n(d) y = sinx + C, C is an arbitary constant
\nHint:
\n\\(\\frac { dy }{dx}\\) = cos x
\ndy = cos x dx
\n\u222bdy = \u222bcos x dx \u21d2 y = sin x + c<\/p>\n

Question 21.
\nA homogeneous differential equation of the form \\(\\frac { dy }{dx}\\) = f(\\(\\frac { y }{x}\\)) can be solved by making substitution.
\n(a) y = v x
\n(b) y = y x
\n(c) x = v y
\n(d) x = v
\nSolution:
\n(a) y = v x<\/p>\n

\"Samacheer<\/p>\n

Question 22.
\nA homogeneous differential equation of the form \\(\\frac { dy }{dx}\\) = f(\\(\\frac { x }{y}\\)) can be solved by making substitution.
\n(a) x = v y
\n(b) y = v x
\n(c) y = v
\n(d) x = v
\nSolution:
\n(a) y = v x<\/p>\n

Question 23.
\nThe variable separable form of \\(\\frac { dy }{dx}\\) = \\(\\frac { y(x-y) }{x(x+y)}\\) by taking y = v x and \\(\\frac { dy }{dx}\\) = v + x \\(\\frac { dy }{dx}\\)
\n(a) \\(\\frac { 2v^2 }{1+v}\\) dv = \\(\\frac { dx }{x}\\)
\n(b) \\(\\frac { 2v^2 }{1+v}\\) dv = –\\(\\frac { dx }{x}\\)
\n(c) \\(\\frac { 2v^2 }{1-v}\\) dv = \\(\\frac { dx }{x}\\)
\n(d) \\(\\frac { 1+v }{2v^2}\\) dv = –\\(\\frac { dx }{x}\\)
\nSolution:
\n(d) \\(\\frac { 1+v }{2v^2}\\) dv = –\\(\\frac { dx }{x}\\)
\nHint:
\n\"Samacheer<\/p>\n

\"Samacheer<\/p>\n

Question 24.
\nWhich of the following is the homogeneous differential equation?
\n(a) (3x – 5) dx = (4y – 1) dy
\n(b) xy dx – (x\u00b3 + y\u00b3) dy = 0
\n(c) y\u00b2dx + (x\u00b2 – xy – y\u00b2) dy = 0
\n(d) (x\u00b2 + y) dx (y\u00b2 + x) dy
\nSolution:
\n(c) y\u00b2dx + (x\u00b2 – xy – y\u00b2) dy = 0<\/p>\n

Question 25.
\nThe solution of the differential equation \\(\\frac { dy }{dx}\\) = \\(\\frac { y }{x}\\) + \\(\\frac { f(\\frac { y }{x}) }{ f(\\frac { y }{x}) }\\) is
\n(a) f\\(\\frac { y }{x}\\) = k x
\n(b) x f\\(\\frac { y }{x}\\) = k
\n(c) f\\(\\frac { y }{x}\\) = k y
\n(d) x f\\(\\frac { y }{x}\\) = k
\nSolution:
\n(a) f\\(\\frac { y }{x}\\) = k x<\/p>\n

\"Samacheer<\/p>\n","protected":false},"excerpt":{"rendered":"

Tamilnadu State Board New Syllabus\u00a0Samacheer Kalvi 12th Business Maths Guide Pdf Chapter 4 Differential Equations Ex 4.6 Text Book Back Questions and Answers, Notes. Tamilnadu Samacheer Kalvi 12th Business Maths Solutions Chapter 4 Differential Equations Ex 4.6 Choose the most suitable answer from the given four alternatives: Question 1. The degree of the differential equation …<\/p>\n","protected":false},"author":3,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":false,"jetpack_social_options":{"image_generator_settings":{"template":"highway","enabled":false},"version":2}},"categories":[5],"tags":[],"class_list":["post-32179","post","type-post","status-publish","format-standard","hentry","category-class-12"],"jetpack_publicize_connections":[],"jetpack_sharing_enabled":true,"jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/samacheerkalvi.guide\/wp-json\/wp\/v2\/posts\/32179"}],"collection":[{"href":"https:\/\/samacheerkalvi.guide\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/samacheerkalvi.guide\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/samacheerkalvi.guide\/wp-json\/wp\/v2\/users\/3"}],"replies":[{"embeddable":true,"href":"https:\/\/samacheerkalvi.guide\/wp-json\/wp\/v2\/comments?post=32179"}],"version-history":[{"count":1,"href":"https:\/\/samacheerkalvi.guide\/wp-json\/wp\/v2\/posts\/32179\/revisions"}],"predecessor-version":[{"id":41575,"href":"https:\/\/samacheerkalvi.guide\/wp-json\/wp\/v2\/posts\/32179\/revisions\/41575"}],"wp:attachment":[{"href":"https:\/\/samacheerkalvi.guide\/wp-json\/wp\/v2\/media?parent=32179"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/samacheerkalvi.guide\/wp-json\/wp\/v2\/categories?post=32179"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/samacheerkalvi.guide\/wp-json\/wp\/v2\/tags?post=32179"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}