Question

Find the complete solution to the following differential equations.

a) y''+2y'-y=10

b) 2y''- y = 3x^2

Answer #1

a) Given differential equation is : y''+2y'-y = 10

i.e., D^{2}y+2Dy-y = 10

i.e., (D^{2}+2D-1)y = 10

Let the trial solution be y = Ae^{mx}.

Then the auxiliary equation is m^{2}+2m-1 = 0 which
gives
.

Therefore the general solution is : ,i.e., , where A, B are constants.

Next we have, for a particular integral,

i.e.,

i.e.,

i.e.,

i.e.,

i.e.,

i.e.,

i.e.,

Therefore, the complete solution is : , where A, B are constants.

b) Given differential equation is : 2y''-y = 3x^{2}

i.e., 2D^{2}y-y = 3x^{2}

i.e., (2D^{2}-1)y = 3x^{2}

Let the trial solution be y = Ae^{mx}.

Then the auxiliary equation is 2m^{2}-1 = 0 which gives
.

Therefore the general solution is : , where A, B are constants.

Next we have, for a particular integral,

i.e.,

i.e.,

i.e.,

i.e.,

i.e.,

i.e.,

i.e.,

Therefore, the complete solution is : , where A, B are constants.

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.

Find the general solution to the differential equation
2y'+y=3x

Find the particular solution of the differential equation: y "+ 2y
'+ y = 3x + 5 + 4e^-x

Find the general solutions of the given systems of differential
equations in the following problem.
x'=3x-2y+et
y'=x

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.

Differential Equations
(a) By inspection, find a particular solution of
y'' + 2y = 14.
yp(x)
= ______ANSWER HERE______.
(b) By inspection, find a particular solution of
y'' + 2y =
−4x.
yp(x)
= ______ANSWER HERE______
(c) Find a particular solution of y'' +
2y = −4x + 14.
yp(x)
= _____ANSWER HERE_____
(d) Find a particular solution of y'' +
2y = 8x + 7.
yp(x)
= ___ANSWER HERE____.

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

Using variation of parameters, find a particular solution of the
given differential equations:
a.) 2y" + 3y' - 2y = 25e-2t (answer should be: y(t) =
2e-2t (2e5/2 t - 5t - 2)
b.) y" - 2y' + 2y = 6 (answer should be: y = 3 + (-3cos(t) +
3sin(t))et )

Find a particular solution for the differential equation by
variation of parameters.
y''- y' -2y = e^3x , y(0) = -3/4 , y'(0)=15/4

Differential Equations problem
If y1= e^-x is a solution of the differential equation
y'''-y''+2y=0 . What is the general solution of the differential
equation?

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