Question

In: Advanced Math

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 a unique twice differentiable solution?

(c) Now suppose that y(0) = y0, y′(0) = v0. Is there a solution in this case (how does it depend on y0 and v0? If so, how many? Explain.

Solutions

Expert Solution


Related Solutions

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
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
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)
The indicated function y1(x) is a solution of the given differential equation. Use reduction of order...
The indicated function y1(x) is a solution of the given differential equation. Use reduction of order to find y2(x) x2y'' − xy' + 17y = 0 ;   y1=xsin(4In(x))
The indicated function y1(x) is a solution of the given differential equation. Use reduction of order...
The indicated function y1(x) is a solution of the given differential equation. Use reduction of order or formula (5) in Section 4.2, y2 = y1(x) e−∫P(x) dx y 2 1 (x) dx (5) as instructed, to find a second solution y2(x). x2y'' − xy' + 26y = 0; y1 = x sin(5 ln(x)).......................................
a) Verify that y1 and y2 are fundamental solutions of the given homogenous second-order linear differential...
a) Verify that y1 and y2 are fundamental solutions of the given homogenous second-order linear differential equation b) find the general solution for the given differential equation c) find a particular solution that satisfies the specified initial conditions for the given differential equation y'' - y = 0 y1 = e^x, y2 = e^-x : y(0) = 0, y'(0) = 5
Find the general solution of the given second-order differential equation. 1. 4?'' + 9? = 15...
Find the general solution of the given second-order differential equation. 1. 4?'' + 9? = 15 2. (1/4) ?'' + ?' + ? = ?2 − 3x Solve the differential equation by variation of parameters. 3. ?'' + ? = sin(x)
1)Find the general solution of the given second-order differential equation. y'' − 7y' + 6y =...
1)Find the general solution of the given second-order differential equation. y'' − 7y' + 6y = 0 2)Solve the given differential equation by undetermined coefficients. y'' + 4y = 6 sin(2x)
Use the method of variation of parameters to find a particular solution of the given differential...
Use the method of variation of parameters to find a particular solution of the given differential equation and then find the general solution of the ODE. y'' + y = tan(t)
For Exercises 1-4 below, (a) verify that y1 and y2 satisfy the given second-order equation, and...
For Exercises 1-4 below, (a) verify that y1 and y2 satisfy the given second-order equation, and (b) find the solution satisfying the given initial conditions (I.C.). 2. y′′−3y′+2y=0; y1(x)=e^x,y2(x)=e^2x. I.C.y(0)=0,y′(0)=−1. 3. y′′−2y′+y=0; y1(x)=e^x,y2(x)=xe^x. I.C.y(0)=1,y′(0)=3.
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT