Differential Equation Analysis

The given equation is an ordinary differential equation (ODE) of the second order:

\[\frac{d^2y}{dx^2}+\frac{dy}{dx}-6y=0\]

This is a homogeneous linear differential equation with constant coefficients. To find its solution, we use the method of characteristic equations.

Characteristic Equation Formulation

For a homogeneous linear differential equation with constant coefficients, we assume a solution of the form \(y = e^{mx}\), where \(m\) is a constant.

Differentiating \(y\) with respect to \(x\):

• First derivative: \(\frac{dy}{dx} = me^{mx}\)
• Second derivative: \(\frac{d^2y}{dx^2} = m^2e^{mx}\)

Substitute these expressions back into the original differential equation:

\[m^2e^{mx} + me^{mx} - 6e^{mx} = 0\]

Since \(e^{mx} \neq 0\) for any finite \(m\) or \(x\), we can divide the entire equation by \(e^{mx}\) to obtain the characteristic equation:

\[m^2 + m - 6 = 0\]

Roots Determination for Characteristic Equation

The characteristic equation is a quadratic equation. We need to find its roots. We can factor this quadratic equation:

We look for two numbers that multiply to -6 and add up to 1 (the coefficient of \(m\)). These numbers are 3 and -2.

So, the equation can be factored as:

\[(m+3)(m-2) = 0\]

Setting each factor to zero gives us the roots:

• \(m+3 = 0 \implies m_1 = -3\)
• \(m-2 = 0 \implies m_2 = 2\)

The roots are real and distinct.

General Solution Derivation

For a second-order homogeneous linear differential equation with constant coefficients, if the characteristic equation has two distinct real roots, say \(m_1\) and \(m_2\), the general solution is given by:

\[y = c_1e^{m_1x} + c_2e^{m_2x}\]

where \(c_1\) and \(c_2\) are arbitrary constants determined by initial conditions (if any).

Substituting the roots \(m_1 = -3\) and \(m_2 = 2\) into the general solution formula:

\[y = c_1e^{-3x} + c_2e^{2x}\]

Solution Verification

Comparing this derived general solution with the given options:

• Option 1: \(y = c_1e^{3x} + c_2e^{-2x}\) (Incorrect)
• Option 2: \(y = c_1e^{3x} + c_2e^{2x}\) (Incorrect)
• Option 3: \(y = c_1e^{-3x} + c_2e^{2x}\) (Correct)
• Option 4: \(y = c_1e^{-3x} + c_2e^{-2x}\) (Incorrect)

The calculated solution matches Option 3.