t2 (d2r/dt2) - 9t (dr/dt) + 16r = 4 is an example of a "Cauchy-Euler equation."...

t² (d²r/dt²) - 9t (dr/dt) + 16r = 4 is an example of a "Cauchy-Euler equation." Such equations appear in a number of physics and engineering applications.

a) Write the complementary homogeneous equation.

b) Plug r = e^kt into the equation you wrote in part a. Show that this solution will not work for any constant k: this equation has no exponential solution.

c) Plug the guess r = tⁿ (where n is a constant) into the equation you wrote for part a. Solve the resulting algebraic equation for n; you should find 2 solutions.

d) Write the general solution to the equation you wrote in Part a.

e) Find a specific solution to the origianl (inhomogeneous) equation.

f) Write the general solution to the inhomogeneous equation.

g) What is it about this particular equation that made r = tⁿ work?

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genius_generous answered 1 year ago

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