Question

The function y_{1}(t) = t is a solution to the
equation.

t^{2} y'' + 2ty' - 2y = 0, t > 0

Find another particular solution y_{2} so that
y_{1} and y_{2} form a fundamental set of
solutions. This means that, after finding a solution y_{2,}
you also need to verify that {y_{1}, y_{2}} is
really a fundamental set of solutions.

Answer #1

Given that y1 = t, y2 = t 2 are solutions to the homogeneous
version of the nonhomogeneous DE below, verify that they form a
fundamental set of solutions. Then, use variation of parameters to
find the general solution y(t).
(t^2)y'' - 2ty' + 2y = 4t^2 t > 0

Two solutions to the diﬀerential equation y00 + 2y0 + y = 0 are
y1(t) = e−t and y2(t) = te−t. Verify that y1(t) is a solution and
show that y1,y2 form a fundamental set of solutions by computing
the Wronskian

The nonhomogeneous equation t2 y′′−2 y=29 t2−1, t>0, has
homogeneous solutions y1(t)=t2, y2(t)=t−1. Find the particular
solution to the nonhomogeneous equation that does not involve any
terms from the homogeneous solution.

Consider the differential equation t 2 y" + 3ty' + y = 0, t >
0. (a) Check that y1(t) = t −1 is a solution to this equation. (b)
Find another solution y2(t) such that y1(t) and y2(t) are linearly
independent (that is, y1(t) and y2(t) form a fundamental set of
solutions for the differential equation)

The nonhomogeneous equation t2 y′′−2 y=19
t2−1, t>0, has homogeneous solutions
y1(t)=t2, y2(t)=t−1. Find the particular
solution to the nonhomogeneous equation that does not involve any
terms from the homogeneous solution.
Enter an exact answer.
Enclose arguments of functions in parentheses. For example,
sin(2x).
y(t)=

Let y1 and y2 be solutions of Bessel's equation t2y" + ty' + (t2
- n2)y =0 on the interval 0 < t < oo, with y1(l)= l, y!(l)=O,
yil)=O, and y2(l)= I.
Compute W[y1,y2](t).

Consider the differential equation
L[y] = y′′ + p(t)y′ + q(t)y = f(t) + g(t), and suppose L[yf] =
f(t) and L[yg] = g(t).
Explain why yp = yf + yg is a solution to L[y] = f + g.
Suppose y and y ̃ are both solutions to L[y] = f + g, and
suppose
{y1, y2} is a fundamental set of solutions to the homogeneous
equation L[y] = 0. Explain why
y = C1y1 + C2y2 + yf...

Find the function y1(t) which is the solution of 4y″+32y′+64y=0
with initial conditions y1(0)=1,y′1(0)=0.
y1(t)=?
Find the function y2(t) which is the solution of 4y″+32y′+64y=0
with initial conditions y2(0)=0, y′2(0)=1.
y2(t)= ?
Find the Wronskian of these two solutions you have found:
W(t)=W(y1,y2).
W(t)=?

Let y1 and y2 be two solutions of the equation y'' + a(t)y' +
b(t)y = 0 and let W(t) = W(y1, y2)(t) be the Wronskian. Determine
an expression for the derivative of the Wronskian with respect to t
as a function of the Wronskian itself.

Consider the differential equation:
66t^2y''+12t(t-11)y'-12(t-11)y=5t^3, . You can verify that y1 = 5t
and y2 = 4te^(-2t/11)satisfy the corresponding homogeneous
equation.
The Wronskian W between y1 and y2 is W(t) =
(-40/11)t^2e^((-2t)/11)
Apply variation of parameters to find a particular solution.
yp = ?????

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