Step 1:

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The differential equation is $\"\"$ .

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All the coefficients are constants and the differential equation is of the form $\"\"$ .

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The auxiliary equation is $\"\"$ .

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The formula for the roots of the quadratic equation $\"\"$ is

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$\"\"$ .

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The roots of auxiliary equation is $\"\"$ are

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$ .

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The roots of auxiliary equation are $\"\"$ and $\"\"$ .

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If the roots of auxiliary equation $\"\"$ are the complex numbers $\"\"$ , $\"\"$ , then the general solution of $\"\"$ is $\"\"$ .

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The roots of auxiliary equation are $\"\"$ and $\"\"$ .

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In this case $\"\"$ .

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The general solution of $\"\"$ is $\"\"$ .