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Chapter 15: Problem 2

Find the general solution of the differential equation. $$y^{\prime \prime}-2 y^{\prime}-6 y=0$$

Short Answer

Expert verified
The general solution of the differential equation is \(y = C_1 e^{3x} + C_2 e^{-2x}\).

Step by step solution

01

Form the Characteristic Equation

The characteristic equation of the given differential equation \(y^{\prime \prime}-2 y^{\prime}-6 y=0\) is found by replacing the second derivative of \(y\) with \(r^2\), the first derivative of \(y\) with \(r\), and \(y\) with 1. This gives the characteristic equation: \[r^2 -2r -6 = 0\]
02

Solve the Characteristic Equation

Now find the roots of the characteristic equation \(r^2 -2r -6 = 0\). This can be done by any suitable method, for example factoring, completing the square, or using the quadratic formula. In this case, we can factor the equation into two factors as follows: \[(r - 3)(r + 2) = 0\] Then the roots are: \[r = 3, -2\]
03

Write Down the General Solution

The general solution to a second-order homogeneous differential equation is written as a linear combination of the exponential of the roots. Therefore our solution is: \[y = C_1 e^{3x} + C_2 e^{-2x}\] where \(C_1\) and \(C_2\) are constants to be determined by any given initial conditions.

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