(a)

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Step 1:

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The differential equation is $\"\"$ .

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The differential equation is in the form of $\"\"$ .

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$\"\"$ is called complementary equation.

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

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The auxiliary equation is $\"\"$ .

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The roots of auxiliary equation is

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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 is real and equal.

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The solution of complementary equation is $\"\"$ .

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Step 2:

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Consider $\"\"$ .

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The $\"\"$ is exponential function and continuous for all values of $\"\"$ .

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

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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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Substitute $\"\"$ in the general solution of $\"\"$ .

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$\"\"$ .

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The solution of differential equation is $\"\"$ .

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

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$\"\"$ .

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Solution:

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$\"\"$ .

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Step 2:

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(b)

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The differential equation is $\"\"$ .

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The differential equation is in the form of $\"\"$ .

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$\"\"$ is called complementary equation.

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

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The auxiliary equation is $\"\"$ .

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The roots of auxiliary equation is

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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 is real and equal.

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The solution of complementary equation is $\"\"$ .

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

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Solving non-homogenous differential equation:

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If the differential equation is in the form of $\"\"$ , then general solution of the non-homogenous differential equation is $\"\"$ , where $\"\"$ is the general solution of the complementary equation and $\"\"$ is the particular solution.

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General solution of the complementary equation:

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If the differential equation is in the form of $\"\"$ , then general solution of the complementary equation is $\"\"$

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Particular solution of the differential equation :

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If the differential equation is in the form of $\"\"$ then the particular solution of the equation is $\"\"$ , where

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$\"\"$ and $\"\"$ .

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Here $\"\"$ is the wronskian of $\"\"$ and $\"\"$ .

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$\"\"$ .

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The particular solution of the differential equation is in the form of $\"\"$ , where $\"\"$ and $\"\"$ .

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Find wronskian of $\"\"$ and $\"\"$ is

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$\"\"$

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Step 3:

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Find $\"\"$ .

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$\"\"$

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$\"\"$

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Find $\"\"$ .

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$\"\"$

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$\"\"$

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Step 4:

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Substitute the values of $\"\"$ , $\"\"$ $\"\"$ and $\"\"$ in $\"\"$ .

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$\"\"$ .

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$\"\"$

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General solution of the differential equation is $\"\"$ .

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$\"\"$

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

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