General Solution of Second Order Differential Equations (Solution of Ex:4.3)

In mathematics, differential equations are used to model real-world phenomena that change over time. However, A second-order differential equation is one in which the highest derivative of the unknown function is a second-order derivative. Therefore, The general solution of a second-order differential equations with constant coefficients is a combination of two functions, one exponential and the other trigonometric.

(Solution of Ex:4.3)

Example of General Solution of Second Order Differential Equations

Firstly, The given second-order differential equation is of the form:

y” + py’ + qy = 0

where p and q are constants.

However, To find the general solution of this differential equation, we assume that the solution is of the form:

y = e^(rt)

where r is a constant. Substituting this into the differential equation, we get:

(r^2 + pr + q)e^(rt) = 0

Since e^(rt) is never zero, we can divide both sides of the equation by e^(rt) to get:

r^2 + pr + q = 0

This is called the characteristic equation of the differential equation. We solve for the roots of this equation, which are given by:

r_1,2 = (-p ± √(p^2 – 4q))/2

There are three cases to consider, depending on the discriminant

$D = p^2 - 4q:$

Case 1: D > 0

$r_1 and r_2$ . distinct real roots.

$y = e^(r_1t)c_1 + e^(r_2t)c_2$

where c_1 and c_2 are constants determined by the initial or boundary conditions.

Case 2: D = 0

$r_1 = r_2 = -p/2.$ repeated root

$y = (c_1 + c_2t)e^(r_1t)$

Case 3: D < 0

$r_1 = α + iβ$ and $r_2 = α - iβ$ , complex conjugate roots

$y = e^(αt)(c_1cos(βt) + c_2sin(βt))$

Conclusion of General Solution of Second Order Differential Equations

In conclusion, the general solution of a 2nd-order differential equation with constant coefficients is a combination of two functions, one exponential and the other trigonometric. Finally, The specific form of the general solution depends on the roots of the characteristic equation, which are determined by the coefficients of the differential equation.

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