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Second Order first degree differential equations with constant coefficients

**Second Order first degree differential
equations with constant coefficients**

**A general second order linear differential
equation with constant coefficients is of the form**

where
ϕ(D) = aD^{2} + bD + c (a,b and c are constants)

To solve the equation (1), we first solve the equation ϕ ( *D* )
*y* = 0. The solution so
obtained is called complementary function (*C.F*).

Next we operate on *f* (
*x*) with 1/ ϕ (* **D*) , the solution so obtained is called particular integral (*P.I*)

**Type 1 : ***f*(*x*) = 0

(i.e) ϕ(D)y = 0

To solve this, put ϕ ( *D*)
= 0

Replace *D* by *m*. This equation is called auxiliary
equation . ϕ( *m* ) = 0 *is a* quadratic
equation. So we have two roots, say *m*_{1} and *m*_{2}.

Now we have the following three cases

Here
A and B are arbitrary constants

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12th Business Maths and Statistics : Chapter 4 : Differential Equations : A general second order linear differential equation with constant coefficients is of the form |

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