{"id":4252,"date":"2023-12-17T06:09:07","date_gmt":"2023-12-17T00:39:07","guid":{"rendered":"https:\/\/wordpress-505192-1602719.cloudwaysapps.com\/?p=4252"},"modified":"2023-12-18T17:09:42","modified_gmt":"2023-12-18T11:39:42","slug":"samacheer-kalvi-12th-maths-solutions-chapter-10-ex-10-5","status":"publish","type":"post","link":"https:\/\/samacheerkalviguru.com\/samacheer-kalvi-12th-maths-solutions-chapter-10-ex-10-5\/","title":{"rendered":"Samacheer Kalvi 12th Maths Solutions Chapter 10 Ordinary Differential Equations Ex 10.5"},"content":{"rendered":"

You can Download\u00a0<\/span>Samacheer Kalvi 12th Maths Book Solutions<\/a>\u00a0Guide Pdf, Tamilnadu State Board help you to revise the complete Syllabus and score more marks in your examinations.<\/span><\/p>\n

Tamilnadu Samacheer Kalvi 12th Maths Book Solutions Chapter 10 Ordinary Differential Equations Ex 10.5<\/h2>\n

Question 1.
\nIf F is the constant force generated by the motor of an automobile of mass M, its velocity V is given by M\\(\\frac{d \\mathbf{V}}{d t}\\) = F – kV, where A is a constant. Express V in terms of t given that V = 0 when t = 0.
\nSolution:
\n\"Samacheer
\nSubstituting in (1)
\n\"Samacheer<\/p>\n

Question 2.
\nThe velocity v, of a parachute falling vertically satisfies the equation, \"Samacheer where g and k are constants. If v and x are both initially zero, find v in terms of x.
\nSolution:
\n\"Samacheer
\n\"Samacheer
\nSquaring on both sides
\n\"Samacheer<\/p>\n

\"Samacheer<\/p>\n

Question 3.
\nFind the equation of the curve whose slope is \\(\\frac{y-1}{x^{2}+x}\\) and which passes through the point (1, 0).
\nSolution:
\n\"Samacheer
\n\"Samacheer
\nThis passes through (1, 0)
\n\"Samacheer<\/p>\n

Question 4.
\nSolve the following differential equations:
\n\"Samacheer
\nSolution:
\n\"Samacheer<\/p>\n

(ii) ydx + (1 + x2<\/sup>) tan-1<\/sup> x dy = 0
\nSolution:
\nydx + (1 + x2<\/sup>) tan-1<\/sup> xdy = 0
\n(1 + x2<\/sup>) tan-1<\/sup>x dy = -y dx
\nSeparating the variables
\n\"Samacheer
\nlog y = – log (tan-1<\/sup> x) + log c
\nlog y + log (tan-1<\/sup> x) = log c
\nlog (y tan-1<\/sup> x) = log c
\ny tan-1<\/sup> x = c<\/p>\n

\"Samacheer<\/p>\n

(iii) sin \\(\\frac{d y}{d x}\\) = a, y (0) = 1
\nSolution:
\n\"Samacheer
\nGiven y(0) = 1
\ni.e., When x = 0, y = 1
\n\u2234 1 = 0(sin-1<\/sup> a) + c \u21d2 c = 1
\n\u2234 The solution is y = xsin-1<\/sup> a + 1
\n\"Samacheer<\/p>\n

(iv) \"Samacheer
\nSolution:
\n\"Samacheer<\/p>\n

(v) (ey<\/sup> + 1) cos x dx + ey<\/sup> sin x dy = 0
\nSolution:
\n(ey<\/sup> + 1) cos x dx + ey<\/sup> sin x dy = 0
\ney<\/sup> sin x dy = – (ey<\/sup> + 1) cos x dx
\n\"Samacheer
\nlog (ey<\/sup> + 1) = – log sin x + log c
\nlog [(ey<\/sup> + 1) + log sin x = log c
\nlog (ey<\/sup> +1) sin x] = log c
\n(ey<\/sup>+ 1) sin x = c<\/p>\n

\"Samacheer<\/p>\n

(vi) \"Samacheer
\nSolution:
\n\"Samacheer<\/p>\n

(vii) \"Samacheer
\nSolution:
\n\"Samacheer<\/p>\n

(viii) x cos y dy = ex<\/sup>(x log x + 1)dx
\nSolution:
\n\"Samacheer<\/p>\n

(ix) \"Samacheer
\nSolution:
\n\"Samacheer<\/p>\n

(x) \"Samacheer
\nSolution:
\n\\(\\frac{d y}{d x}\\) = tan2<\/sup> (x +y)
\nLet u = x + y
\nDifferentiating with respect to ‘x’
\n\"Samacheer
\n\"Samacheer<\/p>\n

Samacheer Kalvi 12th Maths Solutions Chapter 10 Ordinary Differential Equations Ex 10.5 Additional Problems<\/h3>\n

Question 1.
\n\"Samacheer
\nSolution:
\nThe given equation can be written as
\n\"Samacheer
\n\"Samacheer<\/p>\n

\"Samacheer<\/p>\n

Question 2.
\n\"Samacheer
\nSolution:
\nPut x +y = z. Differentiating with respect to x we get
\n\"Samacheer<\/p>\n

Question 3.
\nFind the cubic polynomial in x which attains its maximum value 4 and minimum value 0 at x = -1 and 1 respectively.
\nSolution:
\nLet the cubic polynomial bey = f(x). Since it attains a maximum atx = -1 and a minimum at x = 1.
\n\"Samacheer
\nSeparating the variables we have dy = k (x2<\/sup> – 1) dx
\n\"Samacheer
\n\"Samacheer
\nWhen x = – 1, y = 4 and when x =1,7 = 0
\nSubstituting the equation (1) we have
\n2k + 3c = 12; – 2k + 3c =0
\nOn solving we have k = 3 and c = 2. Substituting these values in (1) we get the required cubic polynomial y = x3<\/sup> – 3x + 2.<\/p>\n

\"Samacheer<\/p>\n

Question 4.
\nThe normal lines to a given curve at each point (x, y) on the curve pass through the point (2, 0). The curve passes through the point (2, 3). Formulate the differential equation representing the problem and hence find the equation of the curve.
\nSolution:
\nSlope of the normal at any point P(x, y) = \\(-\\frac{d x}{d y}\\)
\n\"Samacheer
\nSince the curve passes through (2, 3)
\n\"Samacheer<\/p>\n","protected":false},"excerpt":{"rendered":"

You can Download\u00a0Samacheer Kalvi 12th Maths Book Solutions\u00a0Guide Pdf, Tamilnadu State Board help you to revise the complete Syllabus and score more marks in your examinations. Tamilnadu Samacheer Kalvi 12th Maths Book Solutions Chapter 10 Ordinary Differential Equations Ex 10.5 Question 1. If F is the constant force generated by the motor of an automobile …<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"spay_email":"","footnotes":""},"categories":[8],"tags":[],"jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/samacheerkalviguru.com\/wp-json\/wp\/v2\/posts\/4252"}],"collection":[{"href":"https:\/\/samacheerkalviguru.com\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/samacheerkalviguru.com\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/samacheerkalviguru.com\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/samacheerkalviguru.com\/wp-json\/wp\/v2\/comments?post=4252"}],"version-history":[{"count":1,"href":"https:\/\/samacheerkalviguru.com\/wp-json\/wp\/v2\/posts\/4252\/revisions"}],"predecessor-version":[{"id":60161,"href":"https:\/\/samacheerkalviguru.com\/wp-json\/wp\/v2\/posts\/4252\/revisions\/60161"}],"wp:attachment":[{"href":"https:\/\/samacheerkalviguru.com\/wp-json\/wp\/v2\/media?parent=4252"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/samacheerkalviguru.com\/wp-json\/wp\/v2\/categories?post=4252"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/samacheerkalviguru.com\/wp-json\/wp\/v2\/tags?post=4252"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}