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Problems: Solve the following differential equations, 1) ry'-y=x cos x, with y(z/2)=r. Ans.: y = x(sin x+1).

Problems: Solve the following differential equations, 1) ry'-y=x cos x, with y(z/2)=r. Ans.: y = x(sin x+1).

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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4- Linear differential equations Problems: Solve the following differential equations, 1) ry'-y = x° cos x, with y(r/2)= r. 2) dy = (sin x.cos.x-2y cos.x)dx. 3) xyy'-x'Vy +1=xy +y +x+1. Ans.: y = x(sin x +1). Ans.: 4y-2 sin x(sin x +1) -1= Ce in Ans.: Vy +1 = x(Ce-1). +2,C. -2sin a 4) xydx + (y + 1)(x +1)dy= yVx +1dy. Ans.: Va +1 = y-2++ ye dy 5) x' sin y. =x- dx dy Ans.: x = C- cos y 3x 2.x (In.x-4/3)+ C 6) xdy -Ly+xy'(1- In x) ]dx 0. Ans.: dy 7) +y cot x 5e, with y(r/2)=-4. dx 1-Se Ans.: y: sin x dx 8) (1+ sin y)-[x(sec y+ tan y)-2 ycos y] =0. Ans.: x(sec y + tan y) + y? = C. dy 9) 6y'dr+(3x*y- -4.xy)dy = 0. y 3 Ans.: 10) (1+x*)dy +(xy +x' +x)dx= 0. Ans.: 11) xln xdy = (In.x-y)dx. 12) (2xsin y-tan y)dy+(1-cos y)dx 0. Ans.: x(1-cos y) =cos y-Incos y+ Cc. 13) 2ydx + x(x' In y- 1)dy 0. Ans.: 2y In x = Inx+C. Ans.: y(1+ x -x In y) = Cx. 14) (y+2x'y)dx+ xdy = 0. Ans.: (xy)' =I 6 In x+C 15) 30+x')쑤-2.0y(y"-1). Ans: y'= dx 1+C(I +x*) xe' 16) y(6y -x-1)dx + 2xdy 0. Ans.: y be' +C (cos x+C) 17) (xy'+ 2y) esc x =-Vy. Ans.: y = 4.2 18) (y-x+xy cot x)dx+ xdy 0. Ans.: y =- cot x+ x sin x
Transcribed Image Text:4- Linear differential equations Problems: Solve the following differential equations, 1) ry'-y = x° cos x, with y(r/2)= r. 2) dy = (sin x.cos.x-2y cos.x)dx. 3) xyy'-x'Vy +1=xy +y +x+1. Ans.: y = x(sin x +1). Ans.: 4y-2 sin x(sin x +1) -1= Ce in Ans.: Vy +1 = x(Ce-1). +2,C. -2sin a 4) xydx + (y + 1)(x +1)dy= yVx +1dy. Ans.: Va +1 = y-2++ ye dy 5) x' sin y. =x- dx dy Ans.: x = C- cos y 3x 2.x (In.x-4/3)+ C 6) xdy -Ly+xy'(1- In x) ]dx 0. Ans.: dy 7) +y cot x 5e, with y(r/2)=-4. dx 1-Se Ans.: y: sin x dx 8) (1+ sin y)-[x(sec y+ tan y)-2 ycos y] =0. Ans.: x(sec y + tan y) + y? = C. dy 9) 6y'dr+(3x*y- -4.xy)dy = 0. y 3 Ans.: 10) (1+x*)dy +(xy +x' +x)dx= 0. Ans.: 11) xln xdy = (In.x-y)dx. 12) (2xsin y-tan y)dy+(1-cos y)dx 0. Ans.: x(1-cos y) =cos y-Incos y+ Cc. 13) 2ydx + x(x' In y- 1)dy 0. Ans.: 2y In x = Inx+C. Ans.: y(1+ x -x In y) = Cx. 14) (y+2x'y)dx+ xdy = 0. Ans.: (xy)' =I 6 In x+C 15) 30+x')쑤-2.0y(y"-1). Ans: y'= dx 1+C(I +x*) xe' 16) y(6y -x-1)dx + 2xdy 0. Ans.: y be' +C (cos x+C) 17) (xy'+ 2y) esc x =-Vy. Ans.: y = 4.2 18) (y-x+xy cot x)dx+ xdy 0. Ans.: y =- cot x+ x sin x
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