Introduction

To solve a differential equation of the form 2y'' - 4y' - 3y 0, we

first need to rewrite it in standard form. This allows us to identify the

characteristic equation which we can solve to find the roots that dictate the

general solution.

Revisiting the Standard Form of the Differential Equation

Starting with the given differential equation:

2y'' - 4y' - 3y 0, we can rewrite it as:

y'' - 2y' - frac{3}{2} y 0.

Identification of the Characteristic Equation

The method involves assuming a solution of the form y e^{rt}. By

substituting y e^{rt} into the differential equation, we obtain the

characteristic equation:

r^2 - 2r - frac{3}{2} 0.

Solving the Characteristic Equation

Using the quadratic formula r frac{-b pm sqrt{b^2 - 4ac}}{2a}, where

a 1, b -2, c -frac{3}{2}, we can solve for r.

r frac{-(-2) pm sqrt{(-2)^2 - 4 cdot 1 cdot (-frac{3}{2})}}{2 cdot 1}

frac{2 pm sqrt{4 6}}{2}

frac{2 pm sqrt{10}}{2}

1 pm frac{sqrt{10}}{2}.

The General Solution with Complex Roots

For second-order linear differential equations with complex roots of the

form r alpha pm beta i, the general solution is given by:

y(t) e^{alpha t}(C_1 cos(beta t) C_2 sin(beta t)), where C_1 and C_2 are constants.

Given alpha -1 and beta frac{sqrt{2}}{2}, the general solution is:

y(t) e^{-t}(C_1 cos(frac{sqrt{2}}{2} t) C_2 sin(frac{sqrt{2}}{2} t)).

Alternative Solution Using Euler's Identity

Another approach involves assuming a solution proportional to

y(x) e^{lambda x}. By substituting y e^{lambda x}, we obtain:

2 lambda^2 - 4lambda - 3 0. Solving this quadratic equation, we get:

lambda -1 pm ifrac{sqrt{2}}{2}.

The general solution is then:

y(x) c_1 e^{-x} cos(frac{sqrt{2}}{2} x) c_2 e^{-x} sin(frac{sqrt{2}}{2} x).

Conclusion

The general solution of the given differential equation is a combination of

exponential and trigonometric functions, which helps describe the behavior

of the system under consideration. This method can be applied to a wide range

of linear differential equations with constant coefficients.