use the annihilator method to determinethe form of a particular solution for the given equation
The homogeneous of the given equation is
The solution of the homogeneous is
Then, every solution to the given nonhomogeneous equation also satisfies
is the general solution to this homogeneous equation
Comparing (1) & (2)
Higher-Order Cauchy–Euler Equations. A differential equation that can be expressed in the form
where are constants, is called a homogeneous Cauchy–Euler equation. (The second-order case is discussed in Section 4.7.) Use the substitution to help determine a fundamental solution set for the following Cauchy–Euler equations:
Deflection of a Beam Under Axial Force. A uniform beam under a load and subject to a constant axial force is governed by the differential equation
where is the deflection of the beam, L is the length of the beam, k2 is proportional to the axial force, and q(x) is proportional to the load (see Figure 6.2).
(a) Show that a general solution can be written in the form
(b) Show that the general solution in part (a) can be rewritten in the form
(c) Let q(x)=1 First compute the general solution using the formula in part (a) and then using the formula in part (b). Compare these two general solutions with the general solution
which one would obtain using the method of undetermined coefficients.
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