{"id":22443,"date":"2023-07-25T15:50:12","date_gmt":"2023-07-25T10:20:12","guid":{"rendered":"https:\/\/apboardsolutions.in\/?p=22443"},"modified":"2023-07-27T17:44:09","modified_gmt":"2023-07-27T12:14:09","slug":"inter-2nd-year-maths-2b-differential-equations-solutions-ex-8e","status":"publish","type":"post","link":"https:\/\/apboardsolutions.in\/inter-2nd-year-maths-2b-differential-equations-solutions-ex-8e\/","title":{"rendered":"Inter 2nd Year Maths 2B Differential Equations Solutions Ex 8(e)"},"content":{"rendered":"

Practicing the Intermediate 2nd Year Maths 2B Textbook Solutions<\/a> Inter 2nd Year Maths 2B Differential Equations Solutions Exercise 8(e) will help students to clear their doubts quickly.<\/p>\n

Intermediate 2nd Year Maths 2B Differential Equations Solutions Exercise 8(e)<\/h2>\n

I. Find the I.F. of the following differential equations by transforming them into a linear form.<\/span><\/p>\n

Question 1.
\nx\\(\\frac{dy}{dx}\\) – y = 2x\u00b2 sec\u00b2 2x
\nSolution:
\nx\\(\\frac{dy}{dx}\\) – y = 2x\u00b2 sec\u00b2 2x
\n\\(\\frac{dy}{dx}\\) – \\(\\frac{1}{x}\\) . y = 2x. sec\u00b2 2x
\nThis is linear in x
\nI.F = e\u222b-\\(\\frac{1}{x}\\)dx<\/sup> = e-log x<\/sup> = elog 1\/x<\/sup> = \\(\\frac{1}{x}\\)<\/p>\n

Question 2.
\ny\\(\\frac{dy}{dx}\\) – x = 2y\u00b3
\nSolution:
\ny\\(\\frac{dy}{dx}\\) – x = 2y\u00b3
\n\\(\\frac{dy}{dx}\\) – \\(\\frac{1}{y}\\).x = 2y\u00b2
\nI.F = e\u222b-\\(\\frac{1}{x}\\)dx<\/sup> = e-log y<\/sup> = elog 1\/y<\/sup> = \\(\\frac{1}{y}\\)<\/p>\n

II. Solve the following differential equations.<\/span><\/p>\n

Question 1.
\n\\(\\frac{dy}{dx}\\) + y tan x = cos\u00b3 x
\nSolution:
\n\"Inter
\nSolution is 2y = x cos x + sin x. cos\u00b2 x + c. cos x<\/p>\n

\"Inter<\/p>\n

Question 2.
\n\\(\\frac{dy}{dx}\\) + y sec x = tan x
\nSolution:
\nI.F. = e\u222bsec x dx<\/sup> = elog(secx + tan x)<\/sup> = sec x + tan x
\ny. (sec x + tan x)
\n= \u222btan x (sec x + tan x)dx
\n= \u222b(sec x. tan x + tan\u00b2 x)dx
\n= \u222b(sec x tan x + sec\u00b2 x – 1)dx
\nSolution is
\ny(sec x + tan x) = sec x + tan x – x + c<\/p>\n

Question 3.
\n\\(\\frac{dy}{dx}\\) – y tan x = e sec x. dx
\nSolution:
\nI.F. = e-\u222btan x dx<\/sup> = elog cos x<\/sup> = cos x
\ny. cos x = \u222bex<\/sup>.sec x. cos x dx = \u222b ex<\/sup> dx
\n= ex<\/sup> + c<\/p>\n

Question 4.
\nx\\(\\frac{dy}{dx}\\) + 2y = log x.
\nSolution:
\nI.F. = e\u222b\\(\\frac{2}{x}\\)dx<\/sup> = e2log x<\/sup> = elog x\u00b2<\/sup> = x\u00b2
\nSolution is
\n\"Inter<\/p>\n

Question 5.
\n(1 + x\u00b2)\\(\\frac{dy}{dx}\\) + y = etan-1 x<\/sup>
\nSolution:
\n\"Inter
\n\"Inter<\/p>\n

Question 6.
\n\\(\\frac{dy}{dx}+\\frac{2y}{x}\\) = 2x\u00b2.
\nSolution:
\nI.F. = e\u222b\\(\\frac{2}{x}\\)dx<\/sup> = e2log x<\/sup> = elog x\u00b2<\/sup> = x\u00b2
\ny. x\u00b2 = \u222b2x4<\/sup> dx = \\(\\frac{2x^5}{5}\\) + c<\/p>\n

Question 7.
\n\\(\\frac{dy}{dx}+\\frac{4x}{1+x^2}y=\\frac{1}{(1+x^2)^2}\\)
\nSolution:
\n\"Inter<\/p>\n

Question 8.
\nx\\(\\frac{dy}{dx}\\) + y = (1 + x)ex<\/sup>
\nSolution:
\n\"Inter<\/p>\n

Question 9.
\n\\(\\frac{dy}{dx}+\\frac{3x^2}{1+x^3}y=\\frac{1+x^2}{1+x^3}\\).
\nSolution:
\n\"Inter<\/p>\n

Question 10.
\n\\(\\frac{dy}{dx}\\) – y = -2e-x<\/sup>.
\nSolution:
\nI.F = e\u222b-dx<\/sup> = e-x<\/sup>
\ny. e-x<\/sup> = -2\u222be-2x<\/sup> dx = e-2x<\/sup> + c
\ny = e-x<\/sup> + c. ex<\/sup><\/p>\n

\"Inter<\/p>\n

Question 11.
\n(1 + x\u00b2)\\(\\frac{dy}{dx}\\) + y = tan-1<\/sup> x.
\nSolution:
\n\"Inter
\nPut t = tan-1<\/sup> x so that dt = \\(\\frac{dx}{1+x^2}\\)
\nR.H.S = \u222bt. et<\/sup>dt = t. et<\/sup> – \u222bet<\/sup>dt
\n= t. et<\/sup> – et<\/sup> = et<\/sup>(t – 1)
\nSolution is y. etan-1 x<\/sup> = etan-1 x<\/sup> (tan-1<\/sup>1 x – 1) + c
\ny = tan-1<\/sup> x – 1 + c. e-tan-1 x<\/sup><\/p>\n

Question 12.
\n\\(\\frac{dy}{dx}\\) + y tan x = sin x.
\nSolution:
\nI.F. =e\u222btan x dx<\/sup> = elog sec x<\/sup> = sec x
\ny. sec x = \u222b sin x. sec x dx = \u222btan x dx
\n= log sec x + c<\/p>\n

III. Solve the following differential equations.<\/span><\/p>\n

Question 1.
\ncos x. \\(\\frac{dy}{dx}\\) + y sin x = sec\u00b2 x
\nSolution:
\n\\(\\frac{dy}{dx}\\) + tan x. y = sec\u00b3 x
\nI.F. = e\u222btan x dx<\/sup> = elog sec x<\/sup> = sec x
\ny. sec x = \u222bsec4<\/sup> x dx = \u222b(1 + tan\u00b2 x) sec\u00b2 x
\ndx = tan x + \\(\\frac{\\tan^3x}{3}\\) + c<\/p>\n

Question 2.
\nsec x. dy = (y + sin x) dx.
\nSolution:
\n\\(\\frac{dy}{dx}=\\frac{y+sin x}{sec x}\\) = y cos x + sin x. cos x
\n\\(\\frac{dy}{dx}\\) – y. cos x = sin x. cos x
\nI.F. = e-\u222bcos x dx<\/sup> = e– sin x<\/sup>
\n= y. e-sin x<\/sup> = \u222b e-sin x<\/sup>. sin x. cos x dx ……….. (1)
\nConsider \u222be-sin x<\/sup>. sin x. cos x dx
\nt = – sin x \u21d2 dt = -cos x dx
\n\u222be-sin x<\/sup>. sin x. cos x dx = + \u222bet<\/sup> t dt
\n= t. et<\/sup> – et<\/sup> + c.
\n= e-sin x<\/sup> (- sin x – 1) + c
\ny. e-sin x<\/sup> = – e-sin x<\/sup> (sin x + 1) + c
\nor y = – (sin x + 1) + c. esin x<\/sup>.<\/p>\n

Question 3.
\nx log x.\\(\\frac{dy}{dx}\\) + y = 2 log x.
\nSolution:
\n\"Inter<\/p>\n

Question 4.
\n(x + y + 1)\\(\\frac{dy}{dx}\\) = 1.
\nSolution:
\n\\(\\frac{dy}{dx}\\) = x + y + 1
\n\\(\\frac{dy}{dx}\\) – x = y + 1
\nI.F. = e\u222b-dy<\/sup> e-y<\/sup>
\nx . e-y<\/sup> = \u222be-y<\/sup> (y + 1) dy
\n= -(y + 1). e-y<\/sup> + \u222be-y<\/sup>. dy
\n= -(y + 1) e-y<\/sup> – e-y<\/sup>
\n= -(y + 2) e-y<\/sup>+ c
\nx = -(y + 2) + c. e-y<\/sup><\/p>\n

Question 5.
\nx(x – 1)\\(\\frac{dy}{dx}\\) – y = x\u00b3(x – 1)\u00b3
\nSolution:
\n\\(\\frac{dy}{dx}-\\frac{1}{x(x-1)}\\)y = x\u00b2(x – 1)\u00b2
\n\"Inter<\/p>\n

Question 6.
\n(x + 2y\u00b3)\\(\\frac{dy}{dx}\\) = y
\nSolution:
\n\"Inter
\nSolution is x = y(y\u00b2 + c)<\/p>\n

\"Inter<\/p>\n

Question 7.
\n(1 – x\u00b2)\\(\\frac{dy}{dx}\\) + 2xy = x\\(\\sqrt{1-x^2}\\)
\nSolution:
\n\"Inter<\/p>\n

Question 8.
\nx(x – 1)\\(\\frac{dy}{dx}\\) – (x – 2)y = x\u00b3(2x – 1)
\nSolution:
\n\"Inter
\n\"Inter
\nSolution is y(x – 1) = x\u00b2(x\u00b2 – x + c)<\/p>\n

Question 9.
\n\\(\\frac{dy}{dx}\\)(x\u00b2y\u00b3 + xy) = 1
\nSolution:
\n\\(\\frac{dy}{dx}\\) = xy + x\u00b2y\u00b3
\nThis is Bernoulli’s equation
\n\"Inter
\n\"Inter<\/p>\n

Question 10.
\n\\(\\frac{dy}{dx}\\) + x sin 2y = x\u00b3 cos\u00b2 y
\nSolution:
\n\"Inter
\n\"Inter<\/p>\n

\"Inter<\/p>\n

Question 11.
\ny\u00b2 + (1 – \\(\\frac{1}{y}\\)).\\(\\frac{dy}{dx}\\) = 0
\nSolution:
\n\"Inter
\n\"Inter
\nSolution xy = 1 + y + cy.e1\/y<\/sup><\/p>\n","protected":false},"excerpt":{"rendered":"

Practicing the Intermediate 2nd Year Maths 2B Textbook Solutions Inter 2nd Year Maths 2B Differential Equations Solutions Exercise 8(e) will help students to clear their doubts quickly. Intermediate 2nd Year Maths 2B Differential Equations Solutions Exercise 8(e) I. Find the I.F. of the following differential equations by transforming them into a linear form. Question 1. … Read more<\/a><\/p>\n","protected":false},"author":5,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"spay_email":""},"categories":[15],"tags":[],"jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/apboardsolutions.in\/wp-json\/wp\/v2\/posts\/22443"}],"collection":[{"href":"https:\/\/apboardsolutions.in\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/apboardsolutions.in\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/apboardsolutions.in\/wp-json\/wp\/v2\/users\/5"}],"replies":[{"embeddable":true,"href":"https:\/\/apboardsolutions.in\/wp-json\/wp\/v2\/comments?post=22443"}],"version-history":[{"count":1,"href":"https:\/\/apboardsolutions.in\/wp-json\/wp\/v2\/posts\/22443\/revisions"}],"predecessor-version":[{"id":22444,"href":"https:\/\/apboardsolutions.in\/wp-json\/wp\/v2\/posts\/22443\/revisions\/22444"}],"wp:attachment":[{"href":"https:\/\/apboardsolutions.in\/wp-json\/wp\/v2\/media?parent=22443"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/apboardsolutions.in\/wp-json\/wp\/v2\/categories?post=22443"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/apboardsolutions.in\/wp-json\/wp\/v2\/tags?post=22443"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}