- #1
sunrah
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Homework Statement
solve the following differential equation:
t4x'' - 4t3t' + 6t2x = - 12t - 20
Homework Equations
substitution x(t) = tn
The Attempt at a Solution
this is a Euler equation with the following general solution: x(t) = c1t2 + c2t3 worked out using the above substitution.
The particular solution should be obtainable through variation of constants but I just get a nonsense result:
The wronksian = W = 3c1c2t4 - 2c1c2t4 = c1c2t4
therefore:
[itex]x(t) = - x_{1} \int \frac{x_{2}b(t)}{W} dt + x_{2} \int \frac{x_{1}b(t)}{W}dt = x_{1} \int \frac{c_{2}t^{3}(12t + 20)}{c_{1}c_{2}t^{4}}dt - x_{2} \int \frac{c_{1}t^{2}(12t + 20)}{c_{1}c_{2}t^{4}} dt [/itex]
[itex]x(t) = \frac{x_{1}}{c_{1}} \int (12 + \frac{20}{t})dt - \frac{x_{2}}{c_{2}} \int (\frac{12}{t} + \frac{20}{t^{2}}) dt[/itex]
the integration is trivial but definitely isn't a particular solution!
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