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Because of this, different methods need to be used to solve BVPs. For example, the shooting method (and its variants) or global methods like finite differences, [3] Galerkin methods, [4] or collocation methods are appropriate for that class of problems. The Picard–Lindelöf theorem states that there is a unique solution, provided f is ...
For example, the equation x + y = 2x – 1 is solved for the unknown x by the expression x = y + 1, because substituting y + 1 for x in the equation results in (y + 1) + y = 2(y + 1) – 1, a true statement. It is also possible to take the variable y to be the unknown, and then the equation is solved by y = x – 1.
This says that when rises, there is a substitution effect of / towards good 1. At the same time, the rise in has a negative income effect on good 1's demand, an opposite effect of the same size as the substitution effect, so the net effect is zero. This is a special property of the Cobb-Douglas function.
The same concepts also apply if the price of one good goes up instead of down, with the substitution effect reflecting the change in relative prices and the income effect reflecting the fact the income has been soaked up into additional spending on the retained units of the now-pricier good. For example, consider coffee and tea. If the price of ...
The conjecture is that there is a simple way to tell whether such equations have a finite or infinite number of rational solutions. More specifically, the Millennium Prize version of the conjecture is that, if the elliptic curve E has rank r, then the L-function L(E, s) associated with it vanishes to order r at s = 1.
If only one root, say r 1, is real, then r 2 and r 3 are complex conjugates, which implies that r 2 – r 3 is a purely imaginary number, and thus that (r 2 – r 3) 2 is real and negative. On the other hand, r 1 – r 2 and r 1 – r 3 are complex conjugates, and their product is real and positive. [ 23 ]
Buying a car is an enormous financial decision and can cause some sticker shock. Indeed, Kelley Blue Book noted that the average cost of new cars is now more than $47,000 — up a whopping $6,000 ...
If Walras's law has been satisfied, the optimal solution of the consumer lies at the point where the budget line and optimal indifference curve intersect, this is called the tangency condition. [3] To find this point, differentiate the utility function with respect to x and y to find the marginal utilities, then divide by the respective prices ...