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AleksandrR [38]
2 years ago
7

In this problem you will use undetermined coefficients to solve the nonhomogeneous equation y′′+4y′+3y=8te^−t+6e^−t−(9t+6)

Mathematics
1 answer:
Luden [163]2 years ago
7 0

We're given the ODE,

<em>y''</em> + 4<em>y'</em> + 3<em>y</em> = 8<em>t</em> exp(-<em>t </em>) + 6 exp(-<em>t</em> ) - (9<em>t</em> + 6)

(where I denote exp(<em>x</em>) = <em>eˣ </em>)

First determine the characteristic solution:

<em>y''</em> + 4<em>y'</em> + 3<em>y</em> = 0

has characteristic equation

<em>r</em> ² + 4<em>r</em> + 3 = (<em>r</em> + 1) (<em>r</em> + 3) = 0

with roots at <em>r</em> = -1 and <em>r</em> = -3, so the characteristic solution is

<em>y</em> = <em>C</em>₁ exp(-<em>t</em> ) + <em>C</em>₂ exp(-3<em>t</em> )

For the non-homogeneous equation, assume two ansatz solutions

<em>y</em>₁ = (<em>at</em> ² + <em>bt</em> + <em>c</em>) exp(-<em>t </em>)

and

<em>y</em>₂ = <em>at</em> + <em>b</em>

<em />

• <em>y''</em> + 4<em>y'</em> + 3<em>y</em> = 8<em>t</em> exp(-<em>t </em>) + 6 exp(-<em>t</em> ) … … … [1]

Compute the derivatives of <em>y</em>₁ :

<em>y</em>₁ = (<em>at</em> ² + <em>bt</em> + <em>c</em>) exp(-<em>t </em>)

<em>y</em>₁' = (2<em>at</em> + <em>b</em>) exp(-<em>t </em>) - (<em>at</em> ² + <em>bt</em> + <em>c</em>) exp(-<em>t </em>)

… = (-<em>at</em> ² + (2<em>a</em> - <em>b</em>) <em>t</em> + <em>b</em> - <em>c</em>) exp(-<em>t </em>)

<em>y</em>₁'' = (-2<em>at</em> + 2<em>a</em> - <em>b</em>) exp(-<em>t </em>) - (-<em>at</em> ² + (2<em>a</em> - <em>b</em>) <em>t</em> + <em>b</em> - <em>c</em>) exp(-<em>t </em>)

… = (<em>at</em> ² + (<em>b</em> - 4<em>a</em>) <em>t</em> + 2<em>a</em> - 2<em>b</em> + <em>c</em>) exp(-<em>t</em> )

Substitute them into the ODE [1] to get

→   [(<em>at</em> ² + (<em>b</em> - 4<em>a</em>) <em>t</em> + 2<em>a</em> - 2<em>b</em> + <em>c</em>) + 4 (-<em>at</em> ² + (2<em>a</em> - <em>b</em>) <em>t</em> + <em>b</em> - <em>c</em>) + 3 (<em>at</em> ² + <em>bt</em> + <em>c</em>)] exp(-<em>t</em> ) = 8<em>t</em> exp(-<em>t </em>) + 6 exp(-<em>t</em> )

(<em>at</em> ² + (<em>b</em> - 4<em>a</em>) <em>t</em> + 2<em>a</em> - 2<em>b</em> + <em>c</em>) + 4 (-<em>at</em> ² + (2<em>a</em> - <em>b</em>) <em>t</em> + <em>b</em> - <em>c</em>) + 3 (<em>at</em> ² + <em>bt</em> + <em>c</em>) = 8<em>t</em> + 6

4<em>at</em> + 2<em>a</em> + 2<em>b</em> = 8<em>t</em> + 6

→   4<em>a</em> = 8   and   2<em>a</em> + 2<em>b</em> = 6

→   <em>a</em> = 2   and   <em>b</em> = 1

→   <em>y</em>₁ = (2<em>t</em> ² + <em>t </em>) exp(-<em>t </em>)

(Note that we don't find out anything about <em>c</em>, but that's okay since it would have gotten absorbed into the first characteristic solution exp(-<em>t</em> ) anyway.)

• <em>y''</em> + 4<em>y'</em> + 3<em>y</em> = -(9<em>t</em> + 6) … … … [2]

Compute the derivatives of <em>y</em>₂ :

<em>y</em>₂ = <em>at</em> + <em>b</em>

<em>y</em>₂' = <em>a</em>

<em>y</em>₂'' = 0

Substitute these into [2] :

4<em>a</em> + 3 (<em>at</em> + <em>b</em>) = -9<em>t</em> - 6

3<em>at</em> + 4<em>a</em> + 3<em>b</em> = -9<em>t</em> - 6

→   3<em>a</em> = -9   and   4<em>a</em> + 3<em>b</em> = -6

→   <em>a</em> = -3   and   <em>b</em> = 2

→   <em>y</em>₂ = -3<em>t</em> + 2

Then the general solution to the original ODE is

<em>y(t)</em> = <em>C</em>₁ exp(-<em>t</em> ) + <em>C</em>₂ exp(-3<em>t</em> ) + (2<em>t</em> ² + <em>t </em>) exp(-<em>t </em>) - 3<em>t</em> + 2

Use the initial conditions <em>y</em> (0) = 2 and <em>y'</em> (0) = 2 to solve for <em>C</em>₁ and <em>C</em>₂ :

<em>y</em> (0) = <em>C</em>₁ + <em>C</em>₂ + 2 = 2

→   <em>C</em>₁ + <em>C</em>₂ = 0 … … … [3]

<em>y'(t)</em> = -<em>C</em>₁ exp(-<em>t</em> ) - 3<em>C</em>₂ exp(-3<em>t</em> ) + (-2<em>t</em> ² + 3<em>t</em> + 1) exp(-<em>t </em>) - 3

<em>y'</em> (0) = -<em>C</em>₁ - 3<em>C</em>₂ + 1 - 3 = 2

→   <em>C</em>₁ + 3<em>C</em>₂ = -4 … … … [4]

Solve equations [3] and [4] to get <em>C</em>₁ = 2 and <em>C</em>₂ = -2. Then the particular solution to the initial value problem is

<em>y(t)</em> = -2 exp(-3<em>t</em> ) + (2<em>t</em> ² + <em>t</em> + 2) exp(-<em>t </em>) - 3<em>t</em> + 2

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