(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

solve the following differential equation:

t^{4}x'' - 4t^{3}t' + 6t^{2}x = - 12t - 20

2. Relevant equations

substitution x(t) = t^{n}

3. The attempt at a solution

this is a Euler equation with the following general solution: x(t) = c_{1}t^{2}+ c_{2}t^{3}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 = 3c_{1}c_{2}t^{4}- 2c_{1}c_{2}t^{4}= c_{1}c_{2}t^{4}

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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# Homework Help: Help with variation of constants

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