$\"\"$

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The differential equation is $\"\"$ and the initial conditions are $\"\"$ .

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The auxiliary equation is $\"\"$ .

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Find roots of the auxiliary equation.

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$\"\"$

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The roots of the auxiliary equation $\"\"$ .

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The general solution is $\"\"$

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$\"\"$

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Substitute the initial condition $\"\"$ in equation (1)

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$\"\"$

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Differentiate equation (1) with respect to x.

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$\"\"$

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Substitute the initial condition $\"\"$ in above equation.

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$\"\"$

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$\"\"$

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Subtract equation (2) from equation (3).

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$\"\"$

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Substitute $\"\"$ in equation (2).

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$\"\"$

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Substitute $\"\"$ and $\"\"$ in equation (1).

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$\"\"$

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The general solution is $\"\"$ .

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$\"\"$

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The general solution is $\"\"$ .