$\"\"$

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The differential equation is $\"\"$ and initial condition is $\"\"$ .

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Homogenous differential equation:

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If $\"\"$ is a homogenous differential equation, then to find the solution of the differential equation, we substitute $\"\"$ , where $\"\"$ is differentiable function of $\"\"$ .

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Consider $\"\"$ .

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The degree of $\"\"$ and $\"\"$ is $\"\"$ .

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The differential equation is homogenous differential equation of degree $\"\"$ .

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$\"\"$

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Apply derivative on each side with respect to $\"\"$ .

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$\"\"$

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Apply product rule of differentiation: $\"\"$ .

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$\"\"$ .

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$\"\"$

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$\"\"$ .

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$\"\"$

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Substitute $\"\"$ and $\"\"$ in $\"\"$ .

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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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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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Apply integral on each side.

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$\"\"$

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$\"\"$

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Substitute $\"\"$ .

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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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$\"\"$

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$\"\"$

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Exponentiate each side.

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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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$\"\"$

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The initial condition is $\"\"$ .

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Substitute $\"\"$ and $\"\"$ in $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$ .

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Substitute $\"\"$ in $\"\"$ .

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$\"\"$ , $\"\"$ .

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Solution of the differential equation is $\"\"$ , $\"\"$ .

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$\"\"$

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$\"\"$ , $\"\"$ .