Question

4. For each of the following equations find the general solution.

(a) y" + 2y' + 2y = e^{t}sin(t)

(b) y^{(4)} - 4y" + 4y = e^{t} -
te^{2t}

Answer #1

For each of the following equations, find the general solution
(perferably using imaginary numbers):
y′′ −2y′ −3y=3e2t;
y′′ +2y′ =3+sin(2t);
x2y′′ +7xy′ +5y=x;
x2y′′ +xy′ +4y=sin(lnx)+sin(2lnx);
y′′ +y=tcost.

Solve the following second order differential equations:
(a) Find the general solution of y'' − 2y' = sin(3x) using the
method of undetermined coefficients.
(b) Find the general solution of y'' − 2y'− 3y = te^−t using the
method of variation of parameters.

[Cauchy-Euler equations] For the following equations with the
unknown function y = y(x), find the general solution by changing
the independent variable x to et and re-writing the equation with
the new unknown function v(t) = y(et).
x2y′′ +xy′ +y=0
x2y′′ +xy′ +4y=0
x2y′′ +xy′ −4y=0
x2y′′ −4xy′ −6y=0
x2y′′ +5xy′ +4y=0.

Oridinary Differential equations:
given that y=sinx is a solution of
y(4)+2y'''+11y''+2y'+10y=0,
find the general solution of the DE.

3. Find the general solution to each of the following
differential equations.
(a) y'' - 3y' + 2y = 0
(b) y'' - 10y' = 0
(c) y'' + y' - y = 0
(d) y'' + 2y' + y = 0

3. For each of the following equations find a particular
solution yp(t).
(a) y"+4y= e^(5t)
(b) 4y"+4y'+y = 3t(e^t)
(c) y"+4y'+2y=t^2
(d) y"+9y = cos(3t) +4sin(3t)

Find the general solution of each of the following two
homogeneous 2nd order differential equations:
(a) y''' + 4y'' + 4y' = 0
b) y^(4) − 16y'' = 0

Find the general solution of the following nonhomogeneous
equations:
• y '' − 2y ' + 5y = 25x ^2 + 12
• y ''− 3y ' + 2y = 10 sen 2x
• y '' − 2y ' + y = e^x + e^−2x
• y '' − y = x sen x
• y '' − 6y ' + 9y = 5e^x sen x
• y '' + y ' + y + 1 = sen x +...

4. Find a particular solution, and the general solution to the
associated homogeneous equation, of the following differential
equations:
a) y'-2y=6
b) y'+y=3e-t

For each of the following equations, find the general
solution
y''+y=tcost

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