Find a general solution for the differential equation with x as the independent variable:
The general solution for the differential equation with x as the independent variableis .
The given differential equationis . To solve this equation, we look at its auxiliary equation which is . Observe that -1 is a solution of this equation. So,
To get the other two roots of auxillary equation, we need to solve . We have,
We have m = -1, .From (7) of 328 and (18) of page 330, we conclude that the general solution of the given differential equation is where are arbitrary constants.
The solution of the given differential equation is , where are arbitrary constant.
Hence the final solution is
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.
Constructing Differential Equations. Given three functions that are each three times differentiable and whose Wronskian is never zero on (a, b), show that the equation
is a third-order linear differential equation for which is a fundamental solution set. What is the coefficient of y‴ in this equation?
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