Given a differential equation y\" + 4y = cos (2x) i) Find the homogeneous soluti

Question

Given a differential equation y" + 4y = cos (2x) i) Find the homogeneous solution y_ H. ii) Find a particular solution of non-homogeneous equation and write the general solution of this equation. a) Solve the following Cauchy -Euler's equation x^2y" - 4xy' + y = 0 b) Write the general solution. Given an initial value problem 2 y" - 3y' + y = 0 i) Find the general solution ii) Find the solution of this initial value problem with the initial conditions y(0) = 2, y(1) = 1 a) Find the general solution to the following non-homogeneous differential equation y" - 8y = -x^2 + 2x b) Find a solution that satisfies the following initial conditions y(0) = 1, y'(0) = 0. (Systems of Equations by elimination method) Solve the following system of differential equations and find the general solution

Explanation / Answer

(8)(1) :

Y'' + 4y = cos 2x

homogeneous solution

Y'' + 4 Y = 0

auxillary equation

D2 + 4 = 0

D = 2 i , -2i

homogeneous solution

Y = A cos 2x + B sin 2x

(8) ( 2) Non homogeneous solution :

Y'' + 4 Y = cos 2x

C.F. = A cos 2x + B sin 2x

Now we find P.I.

1 / ( D2 + 4) cos 2x

= - x /4 cos 2x

general non homogeneous solution

Y = A cos 2x + B sin 2x - x /4 cos 2x

(9 a) :

X2 Y'' - 4 XY' + Y = 0

euler - cauchy equation

D( D-1) - 4 D + 1 = 0

D2 - D - 4 D + 1 = 0

D2 - 5 D + 1= 0

D = 5 + (21)1/2 / 2 , 5 - (21)1/2 / 2

Y = A X{( 5+ (21)^1/2 )/2}x + B X{ ( 5 - ( 21)^1/2) / 2}x

(9 b) :

general solution

Y = A X{( 5+ (21)^1/2 )/2}x + B X{ ( 5 - ( 21)^1/2) / 2}x

(10 a) :

2 Y''+ 3 y' + Y = 0

auxillary equation

2 D2 + 3 D + 1 = 0

( 2 D - 1) ( D - 1) = 0

solution Y = A ex + B e1/2x

(10 b ) initial condition

Y ( 0 ) = 2 , Y' ( 1) = 1

2 = A + B .........(1)

y ' = A ex + B / 2 e1/2x

1 = A e + B /2 e1/2

now solving this using (1) and we get '

A = 2 - 2( 2e - 1) / (2e - e1/2 )

B = 2 ( 2e-1) / ( 2e - e1/2 )

put these values in (*)

Y =  2 - 2( 2e - 1) / (2e - e1/2 ) ex + 2 ( 2e-1) / ( 2e - e1/2 ) e1/2 x

(11 a) :

Y'' - 8 Y = - X2 + 2 X

D2 - 8 = 0

D = 2 (2)1/2 , -2 (2)1/2

C.F. = A e2(2)^1/2 x + B e-2(2)^1/2 x

P.I. =

1 / D2 - 8 ( - X2 + 2X )

- 1/8 ( 1 - D2 / 8) ( - X2 + 2X)

- 1/8 ( - X2 + 2 X + 1/8)

Y = A e2(2)^1/2 x + B e-2(2)^1/2 x - 1/8 ( - X2 + 2 X + 1/8)

Related Questions

Navigate

• Browse (All)
• Browse G
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.