Use ‘Reduction of Order’ to find a second solution y2 to the given ODEs: (a) y′′+2y′+y=0,...

Use ‘Reduction of Order’ to find a second solution y2 to the given ODEs:

(a) y′′+2y′+y=0, y1 =xe−x

(b) y′′+9y=0, y1 =sin3x
(c) x2y′′+2xy′−6y=0, y1 =x2

(d) xy′′ +y′ =0, y1 =lnx

Expert Solution

Colby Messinger answered 2 years ago

Use the method of reduction of order to find a second solution y2 of the given...

Use the method of reduction of order to find a second solution y2 of the given differential equation such that {y1, y2} is a fundamental set of solutions on the given interval. t2y′′ +2ty′ −2y=0, t > 0, y1(t)=t (a) Verify that the two solutions that you have obtained are linearly independent. (b) Let y(1) = y0, y′(1) = v0. Solve the initial value problem. What is the longest interval on which the initial value problem is certain to have...

Use the method of reduction of order to find a second independent solution of the given...

Use the method of reduction of order to find a second independent solution of the given differential equation. t2y'' + 4ty' − 4y = 0, t > 0; y1(t) = t

Assuming that x>0, use the method of reduction of order to find a second solution to...

Assuming that x>0, use the method of reduction of order to find a second solution to x^2y''−3xy'+4y=0 Given y1(x)=x^2

Find solutions to the following ODEs: • y¨ − y˙ − 2y = t, y(0) =...

Find solutions to the following ODEs: • y¨ − y˙ − 2y = t, y(0) = 0, y˙(0) = 1 • y¨ − 2 ˙y + y = 4 sin(t), y(0) = 1, y˙(0) = 0 • y¨ = t 2 + t + 1 (find general solution only) • y¨ + 4y = t − 2 sin(2t), y(π) = 0, y˙(π) = 1

Question 1. Find the general solution of the following ODEs a) y′′ − 2y′ − 3y...

Question 1. Find the general solution of the following ODEs a) y′′ − 2y′ − 3y = e−x + 2e3x b) y′′ + y′ + y = x2 + 4 cos x.

Find the solution of the given initial value problem. 2y''+y'-4y=0 ; y(0)=0 y'(0)=1

find the solution of the given initial value problem. y′′−2y′−3y=3te^2t y(0)=1 y′(0)=0

Problem 3: Find a second solution by reduction of order - nonhomogeneous The given function y1(x)...

Problem 3: Find a second solution by reduction of order - nonhomogeneous The given function y1(x) is a solution of the associated homogeneous equation. Use the reduction of order method to ﬁnd a solution y(x) = u(x)y1(x) of the nonhomogeneous equation. 1. x^2y'' + xy'−4y = x^3, y1 = x^2 2. 2x^2*y'' + 3xy'−y = 1/x , y1 = x^(1/2)

Find the solution of the equation given around x_0 = 0. (1+x^2)y"+2xy'-2y=0

Find the solution. y'' + 2y' + 0.75y = 2cosx - 0.25sinx + 0.09x , y(0)...

Find the solution. y'' + 2y' + 0.75y = 2cosx - 0.25sinx + 0.09x , y(0) = 2.78 , y'(0) = -0.43

Question