{"id":20961,"date":"2020-01-23T12:49:32","date_gmt":"2020-01-23T07:19:32","guid":{"rendered":"https:\/\/kseebsolutions.guru\/?p=20961"},"modified":"2021-07-02T15:27:17","modified_gmt":"2021-07-02T09:57:17","slug":"2nd-puc-maths-question-bank-chapter-9-miscellaneous-ex","status":"publish","type":"post","link":"https:\/\/kseebsolutions.guru\/2nd-puc-maths-question-bank-chapter-9-miscellaneous-ex\/","title":{"rendered":"2nd PUC Maths Question Bank Chapter 9 Differential Equations Miscellaneous Exercise"},"content":{"rendered":"

Students can Download Maths Chapter 9 Differential Equations Miscellaneous Exercise Questions and Answers, Notes Pdf, 2nd PUC Maths Question Bank with Answers<\/a>\u00a0helps you to revise the complete Karnataka State Board Syllabus and score more marks in your examinations.<\/p>\n

Karnataka 2nd PUC Maths Question Bank Chapter 9 Differential Equations Miscellaneous Exercise<\/h2>\n

Question 1.
\nFor each of the differential equations given below, indicate its order and degree
\n(if defined).<\/p>\n

(i)
\n\\(\\cfrac{d^{2} y}{d x^{2}}+5 x\\left(\\cfrac{d y}{d x}\\right)^{2}-6 y=\\log x\\)
\nAnswer:
\nOrder -2, degree – 1<\/p>\n

(ii)
\n\\(\\left(\\cfrac{d y}{d x}\\right)^{3}-4\\left(\\cfrac{d y}{d x}\\right)^{2}+7 y=\\sin x\\)
\nAnswer:
\nOrder is 1 degree is 3.<\/p>\n

(iii)
\n\\(\\cfrac{d^{4} y}{d x^{4}}-\\sin \\left(\\cfrac{d^{3} y}{d x^{3}}\\right)=0\\)
\nAnswer:
\nOrder is 4. As the equation is not a polynomial in \\(\\frac{d y}{d x}\\), degree is not defined.<\/p>\n

\"KSEEB<\/p>\n

Question 2.
\nFor each of the exercises given below verify that the given function (implicit or explicit) is a solution of the corresponding differential equation.<\/p>\n

(i) y = a ex<\/sup> + b e-x<\/sup> + x2
\n<\/sup>\\(x \\cfrac{d^{2} y}{d x^{2}}+2 \\cfrac{d y}{d x}-x y+x^{2}-2=0\\)
\nAnswer:
\n\"2nd<\/p>\n

(ii) y = ex<\/sup> (a cos x + b sin x)
\n\\(\\cfrac{d^{2} y}{d x^{2}}-2 \\cfrac{d y}{d x}+2 y=0\\)
\nAnswer:
\n\\(\\frac{d y}{d x}\\) = ex<\/sup> (-a sin x + b cos x) + (+a cos x + b sin x) ex
\n\"2nd
\n<\/sup><\/p>\n

(iii) \\(y=x \\sin 3 x \\quad: \\cfrac{d^{2} y}{d x^{2}}+9 y-6 \\cos 3 x=0\\)
\nAnswer:
\n\"2nd<\/p>\n

\"KSEEB<\/p>\n

(iv) \\(2 x=2 y^{2} \\times \\cfrac{1}{y} \\times \\cfrac{d y}{d x}+\\log y \\times 4 y \\times \\cfrac{d y}{d x}\\)
\nAnswer:
\n\"2nd<\/p>\n

Question 3.
\nForm the differential equation representing the family of curves given by
\n(x – a)2<\/sup> + 2y2<\/sup> = a2<\/sup>, where a is an arbitrary constant.
\nAnswer:
\n(x – a)2<\/sup> + 2y2<\/sup> = a2
\n<\/sup>2(x -a ) +2(2y) x \\(\\frac{d y}{d x}\\) = 0
\n(x – a) + 2yx y’ = 0, 2y y’ = – (x – a) = a – x ………….(1)
\n4y2<\/sup> (y’ )2<\/sup> + 2 y2<\/sup> = 4y2<\/sup> (y’ )2<\/sup> + 4 xy y’ + x2
\n<\/sup>\\(4 x y \\frac{d y}{d x}+x^{2}-2 y^{2}=0\\)<\/p>\n

Question 4.
\nProve that x2<\/sup> – y2<\/sup> = c (x2<\/sup> + y2<\/sup>) is the general solution of differential equation (x3<\/sup> – 3x y2<\/sup>) dx = (y3<\/sup> – 3x2<\/sup>y) dy, where c is a parameter.
\nAnswer:
\nx2<\/sup> – y2<\/sup> = c (x2<\/sup> + y2<\/sup>)2
\n\"2nd
\n<\/sup><\/p>\n

\"KSEEB<\/p>\n

Question 5.
\nForm the differential equation of the family of circles in the first quadrant which touch the coordinate axes.
\nAnswer:
\n(x – h)2<\/sup> + (y – h)2<\/sup> = h2 <\/sup>dy
\n2(x – h) +2(y – h) \\(\\frac{d y}{d x}=0\\)
\n\"2nd
\n\"2nd<\/p>\n

Question 6.
\nFind the general solution of the differential equation
\n\\(\\cfrac{d y}{d x}+\\sqrt{\\cfrac{1-y^{2}}{1-x^{2}}}=0\\)
\nAnswer:
\n\"2nd<\/p>\n

Question 7.
\nShow that the general solution of the different equation
\n\\(\\frac{d y}{d x}+\\frac{y^{2}+y+1}{x^{2}+x+1}=0\\) is given by
\n(x +y+1) = A (1- x – y -2xy)
\nAnswer:
\n\"2nd
\n\"2nd<\/p>\n

\"KSEEB<\/p>\n

Question 8.
\nFind the equation of the curve passing through the point \\(\\left(0, \\frac{\\pi}{4}\\right)\\) whose differencial\u00a0 equation is sin x y dy = 0
\nAnswer:
\nSin x cos y dx = – cos x sin y dy
\n\"2nd<\/p>\n

Question 9.
\nFind the particular solution of the differential equation (1 + e2x<\/sup>) dy + (1 + y2<\/sup>) ex<\/sup> dx = 0, given that y = 1 when x = 0.
\nAnswer:
\n\"2nd<\/p>\n

Question 10.
\nSolve the differential equation
\n\\(\\mathbf{y}^{\\mathbf{e}^{\\frac{x}{y}}} \\mathbf{d x}=\\left(\\mathbf{x e}^{\\frac{\\mathbf{x}}{y}}+\\mathbf{y}^{2}\\right) \\mathbf{d y}(\\mathbf{y} \\neq \\mathbf{0})\\)
\nAnswer:
\n\"2nd<\/p>\n

Question 11.
\nFind a particular solution of the differential equation (x – y) (dx + dy) = dx – dy, given that y = -1, when x = 0. (Hint: put x – y = t)
\nAnswer:
\n\"2nd
\nx + y = log (x – y) + C
\nwhen x = 0, y = -1
\n0 – 1 = log(0 + 1) + C
\n\u21d2 C = -1 solution is x + y = log (x – y) – 1.<\/p>\n

\"KSEEB<\/p>\n

Question 12.
\nSolve the differential equation
\n\\(\\left[\\cfrac{\\mathrm{e}^{-2 \\sqrt{x}}}{\\sqrt{x}}-\\cfrac{y}{\\sqrt{x}}\\right] \\cfrac{d x}{d y}=1(x \\neq 0)\\)
\nAnswer:
\n\"2nd<\/p>\n

Question 13.
\nFind a particular solution of the differential dy equation \\(\\frac{d y}{d x}+y \\cot x=4 x \\csc x(x \\neq 0)\\) given that y = 0 when \\(x=\\frac{\\pi}{2} \\)
\nAnswer:
\n\"2nd<\/p>\n

Question 14.
\nFind a particular solution of the differential dy equation (x+1) \\frac{d y}{d x}=2 e^{-y}-1, given that y = 0 when x = 0.
\nAnswer:
\n\"2nd<\/p>\n

\"KSEEB<\/p>\n

Question 15.
\nThe population of a village increases continuously at the rate proportional to the number of its inhabitants present at any time. If the population of the village was 20,000 in 1999 and 25000 in the year 2004, what will be the population of the village in 2009?
\nAnswer:
\n\"2nd
\n\"2nd
\n\"2nd<\/p>\n

Question 16.
\nThe general solution of the differential equation \\(\\frac{y d x-x d y}{y}=0\\) is
\n(A) xy = C
\n(B) x = Cy2
\n<\/sup>(C) y = Cx
\n(D)\u00a0y = Cx2
\n<\/sup>Answer:
\n\"2nd<\/p>\n

Question 17.
\nThe general solution of a differential equation of the type \\(\\frac{d x}{d y}+P_{1} x=Q_{1} \\text { is }\\)
\n\"2nd
\nAnswer:
\n\"2nd<\/p>\n

\"KSEEB<\/p>\n

Question 18.
\nThe general solution of the differential equation ex<\/sup> dy + (y ex<\/sup> + 2x) dx = 0 is
\n(A) x ey<\/sup>+ x2<\/sup> = C
\n(B) x ey<\/sup> + y2<\/sup> = C
\n(C) y ex<\/sup> + x2<\/sup> = C
\n(D) y ey<\/sup> + x2<\/sup> = C
\nAnswer:
\n\"2nd<\/p>\n

2nd PUC Maths Differential Equations Miscellaneous Exercise Additional Questions and Answers<\/h3>\n

Question 1.
\nSolve the differential equations
\n\\(\\sqrt{1+x^{2}+y^{2}+x^{2} y^{2}}+x y \\cfrac{d y}{d x}=0\\) (CBSE 2010)
\nAnswer:
\n\"2nd<\/p>\n

Question 2.
\nsolve
\n\\(\\left(x^{2}-1\\right) \\cfrac{d y}{d x}+2 x y=\\cfrac{1}{x^{2}-1}\\)(CBSE 2010)
\nAnswer:
\n\"2nd
\n\"2nd<\/p>\n

Question 3.
\nShow that \\((x-y) \\frac{d y}{d x}=x+2 y\\) is homogenous and solve it (CBSE 2010)
\nAnswer:
\n\"2nd
\n\"2nd<\/p>\n

\"KSEEB<\/p>\n

Question 4.
\nSolve
\n\\(y d x+x \\log \\left(\\cfrac{y}{x}\\right) d y-2 x d y=0\\) (CBSE 2010)
\nAnswer:
\n\"2nd
\nQuestion 5.
\nsolve
\n\\(\\left(x^{3}+x^{2}+x+1\\right)\\left(\\cfrac{d y}{d x}\\right)=2 x^{2}+x\\) (CBSE 2010)
\nAnswer:
\n\"2nd<\/p>\n

Question 6.
\nSolve
\n\\(\\left(x^{2}+1\\right) \\cfrac{d y}{d x}+2 x y=\\sqrt{x^{2}+4}\\)(CBSE 2010)
\nAnswer:
\n\"2nd
\n\"2nd<\/p>\n

Question 7.
\nSolve \\(x d y-y d x=\\sqrt{x^{2}+y^{2}} d x\\)(CBSE 2011)
\nAnswer:<\/p>\n

\"2nd<\/p>\n

\"KSEEB<\/p>\n

Question 8.
\nSolve
\n\\(\\left(y+3 x^{2}\\right) \\frac{d x}{d y}=x\\)
\nAnswer:
\n\"2nd
\n\"2nd<\/p>\n

Question 9.
\nSolve x dy + (y – x3<\/sup>) dx = 0 (CBSE 2011)
\nAnswer:
\n\"2nd<\/p>\n

Question 10.
\nForm the differential equation of the firmly of circles in the second quadrant and touching the co-ordinate axes. (CBSE 2012)
\nAnswer:
\nCentre is (-h, h), v = h
\n\"2nd
\nequation of circle is (x + h)2<\/sup> + (y – h)2<\/sup> = h2<\/sup>
\n(x – h)2<\/sup> + (y – h)2<\/sup> = h2 <\/sup>dy
\n2(x – h) +2(y – h) \\(\\frac{d y}{d x}=0\\)
\n\"2nd
\n\"2nd<\/p>\n

\"KSEEB<\/p>\n

Question 11.
\n(1 – x)2<\/sup> dy + 2 xy dx = cot x dx
\nAnswer:
\n\"2nd<\/p>\n

Question 12.
\n\\(\\int x\\left(x^{2}-1\\right) \\frac{d x}{d y}=1, y=0, \\text { when } x=2\\) (CBSE 2012)
\nAnswer:
\n\"2nd<\/p>\n

Question 13.
\nFind the particular solution: (CBSE 2012)
\n\\(x \\cfrac{d y}{d x}-y+x \\sin \\left(\\cfrac{y}{x}\\right)=0, \\text { when } x=2, y=\\pi\\)
\nAnswer:
\n\"2nd
\n\"2nd<\/p>\n","protected":false},"excerpt":{"rendered":"

Students can Download Maths Chapter 9 Differential Equations Miscellaneous Exercise Questions and Answers, Notes Pdf, 2nd PUC Maths Question Bank with Answers\u00a0helps you to revise the complete Karnataka State Board Syllabus and score more marks in your examinations. Karnataka 2nd PUC Maths Question Bank Chapter 9 Differential Equations Miscellaneous Exercise Question 1. For each of …<\/p>\n","protected":false},"author":5,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[82],"tags":[],"jetpack_sharing_enabled":true,"jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/kseebsolutions.guru\/wp-json\/wp\/v2\/posts\/20961"}],"collection":[{"href":"https:\/\/kseebsolutions.guru\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/kseebsolutions.guru\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/kseebsolutions.guru\/wp-json\/wp\/v2\/users\/5"}],"replies":[{"embeddable":true,"href":"https:\/\/kseebsolutions.guru\/wp-json\/wp\/v2\/comments?post=20961"}],"version-history":[{"count":0,"href":"https:\/\/kseebsolutions.guru\/wp-json\/wp\/v2\/posts\/20961\/revisions"}],"wp:attachment":[{"href":"https:\/\/kseebsolutions.guru\/wp-json\/wp\/v2\/media?parent=20961"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/kseebsolutions.guru\/wp-json\/wp\/v2\/categories?post=20961"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/kseebsolutions.guru\/wp-json\/wp\/v2\/tags?post=20961"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}