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Obtain a particular integral of the equation: \[ \frac{d^{2} y}{d x^{2}}-6 \fra...

May 22, 2024

Solution by Steps

step 1

We start with the given second-order linear differential equation:

\frac{d^{2} y}{d x^{2}} - 6 \frac{d y}{d x} + 8 y = x

step 2

To find the particular solution, we first solve the corresponding homogeneous equation:

\frac{d^{2} y}{d x^{2}} - 6 \frac{d y}{d x} + 8 y = 0

step 3

The characteristic equation for the homogeneous part is:

r^2 - 6r + 8 = 0

step 4

Solving the characteristic equation, we get the roots:

r = 2 \quad \text{and} \quad r = 4

step 5

Therefore, the general solution to the homogeneous equation is:

y_h(x) = c_1 e^{2x} + c_2 e^{4x}

step 6

Next, we find a particular solution to the non-homogeneous equation. We assume a particular solution of the form:

y_p(x) = Ax + B

step 7

Substituting

y_p(x) = Ax + B

into the original equation, we get:

\frac{d^{2} (Ax + B)}{d x^{2}} - 6 \frac{d (Ax + B)}{d x} + 8 (Ax + B) = x

step 8

Simplifying, we find:

0 - 6A + 8(Ax + B) = x

8Ax + 8B - 6A = x

step 9

Equating coefficients, we get:

8A = 1 \quad \text{and} \quad 8B - 6A = 0

step 10

Solving these equations, we find:

A = \frac{1}{8} \quad \text{and} \quad B = \frac{3}{32}

step 11

Therefore, the particular solution is:

y_p(x) = \frac{x}{8} + \frac{3}{32}

step 12

The general solution to the original differential equation is:

y(x) = y_h(x) + y_p(x) = c_1 e^{2x} + c_2 e^{4x} + \frac{x}{8} + \frac{3}{32}

Answer

The particular integral of the equation is

y(x) = c_1 e^{2x} + c_2 e^{4x} + \frac{x}{8} + \frac{3}{32}

Key Concept

Particular Integral of a Differential Equation

Explanation

To find the particular integral of a second-order linear differential equation, we solve the corresponding homogeneous equation and then find a particular solution to the non-homogeneous equation. The general solution is the sum of these two solutions.