Solving Second Order Linear Homogeneous Differential Equations

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Any differential equation of the formis a second order differential equations and there is a standard technique for solving any equation of this sort. We assume a solution of the formand substitute this into the equation. We extract the non zero factor – since no exponential is zero for any finite x –to obtain a quadratic equation. We solve this equation obtain solutionsandand then the general solution isandmay be evaluated given suitable boundary conditions, for example

Example: Solve the equation(1) subject toandat

Substitution of the above expressions into (1) gives

We can factor out the nonzeroto obtain

Becauseis non zero we can divide by it to obtainand we factorise this expression to obtainand solve to obtainandThe general solution is then

We now have to find

whenimplies (2)

atimplies (3)

(2)+(3) givesthen from (2)

The solution is

Example: Solve the equation(1) subject towhenandat

Substitution of the above expressions into (1) gives

We can moveto the left hand side and factor out the nonzeroto obtain

Becauseis non zero we can divide by it to obtainand we factorise this expression to obtainand solve to obtainandThe general solution is then

We now have to find

whenimplies (2)

atimplies (3)

(2)-(3) givesthen from (2)

The solution is

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