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sdas [7]
2 years ago
12

Solve the following differential equations using classical methods. Assume zero initial conditions.

Mathematics
1 answer:
MA_775_DIABLO [31]2 years ago
7 0

I'll use the integrating factor method for the first DE, and undetermined coefficients for the second one.

(a) Multiply both sides by exp(7<em>t</em> ):

exp(7<em>t</em> ) d<em>x</em>/d<em>t</em> + 7 exp(7<em>t</em> ) <em>x</em> = 5 exp(7<em>t</em> ) cos(2<em>t</em> )

The left side is now the derivative of a product:

d/d<em>t</em> [exp(7<em>t</em> ) <em>x</em>] = 5 exp(7<em>t</em> ) cos(2<em>t</em> )

Integrate both sides:

exp(7<em>t</em> ) <em>x</em> = 10/53 exp(7<em>t</em> ) sin(2<em>t</em> ) + 35/53 exp(7<em>t</em> ) cos(2<em>t</em> ) + <em>C</em>

Solve for <em>x</em> :

<em>x</em> = 10/53 sin(2<em>t</em> ) + 35/53 cos(2<em>t</em> ) + <em>C</em> exp(-7<em>t</em> )

(b) Solve the corresonding homogeneous DE:

d²<em>x</em>/d<em>t</em> ² + 6 d<em>x</em>/d<em>t</em> + 8<em>x</em> = 0

has characteristic equation

<em>r</em> ² + 6<em>r</em> + 8 = (<em>r</em> + 4) (<em>r</em> + 2) = 0

with roots at <em>r</em> = -4 and <em>r</em> = -2. So the characteristic solution is

<em>x</em> (char.) = <em>C₁</em> exp(-4<em>t</em> ) + <em>C₂</em> exp(-2<em>t</em> )

For the particular solution, assume an <em>ansatz</em> of the form

<em>x</em> (part.) = <em>a</em> cos(3<em>t</em> ) + <em>b</em> sin(3<em>t</em> )

with derivatives

d<em>x</em>/d<em>t</em> = -3<em>a</em> sin(3<em>t</em> ) + 3<em>b</em> cos(3<em>t</em> )

d²<em>x</em>/d<em>t</em> ² = -9<em>a</em> cos(3<em>t</em> ) - 9<em>b</em> sin(3<em>t</em> )

Substitute these into the non-homogeneous DE and solve for the coefficients:

(-9<em>a</em> cos(3<em>t</em> ) - 9<em>b</em> sin(3<em>t</em> ))

… + 6 (-3<em>a</em> sin(3<em>t</em> ) + 3<em>b</em> cos(3<em>t</em> ))

… + 8 (<em>a</em> cos(3<em>t</em> ) + <em>b</em> sin(3<em>t</em> ))

= (-<em>a</em> + 18<em>b</em>) cos(3<em>t</em> ) + (-18<em>a</em> - <em>b</em>) sin(3<em>t</em> ) = 5 sin(3<em>t</em> )

So we have

-<em>a</em> + 18<em>b</em> = 0

-18<em>a</em> - <em>b</em> = 5

==>   <em>a</em> = -18/65 and <em>b</em> = -1/65

so that the particular solution is

<em>x</em> (part.) = -18/65 cos(3<em>t</em> ) - 1/65 sin(3<em>t</em> )

and thus the general solution is

<em>x</em> (gen.) = <em>x</em> (char.) + <em>x</em> (part.)

<em>x</em> = <em>C₁</em> exp(-4<em>t</em> ) + <em>C₂</em> exp(-2<em>t</em> ) - 18/65 cos(3<em>t</em> ) - 1/65 sin(3<em>t</em> )

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