Abel gave a new derivation of this for the particular integral he considered and found
2.
Then, using a superposition principle, we can say that the particular integral y _ p is
3.
Then, for the second one, I haven't got a clue what to try for the particular integral.
4.
And a particular integral y _ p of the linear non-homogeneous ordinary differential equation based on g ( x ).
5.
In order to find the particular integral, we need to'guess'its form, with some coefficients left as variables to be solved for.
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Thom's construction thus also unifies differential topology and stable homotopy theory, and is in particular integral to our knowledge of the stable homotopy groups of spheres.
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One reason for the first convention is that the integrability of on an interval implies that is integrable on any subinterval, but in particular integrals have the property that:
8.
A somewhat non-standard version of the tangent half-angle substitution, which is simpler in the case of this particular integral, published in 2013, is as follows:
9.
If the function of " x " is a sum of terms in the above table, the particular integral can be guessed using a sum of the corresponding terms for " y ".
10.
If a term in the above particular integral for y appears in the homogeneous solution, it is necessary to multiply by a sufficiently large power of " x " in order to make the solution independent.
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