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Hi everyone.
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Today we're looking at following differential equation and we're trying to solve it given that we know the fundamental solutions to the homogenous form of this equation where we set the right hand side equal to zero using the method of variation of parameters.
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Now the first step to use the method of variation of parameters is to make sure that this leading term in our equation is only, it only has a one here.
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We have a problem here because there's an x cubed here, but we can get rid of that by dividing our, everything in our differential equation by x cubed.
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If we do that, we simplify a differential equation to a y prime prime, prime, minus two times x inverse times y prime, plus three times x to the negative two times y prime minus three, x to the negative three, y is equal to x to the negative one.
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Now we've done this.
00:58
We write our solution as y equal to v1 times x plus v2 times x log x plus v3 times x cubed, where v1, v2, v3 are functions which satisfy the following matrix equation.
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We write our fundamental solutions as the top row in a square matrix...