Partial differential equations;: An introduction by Eutiquio C Young
By Eutiquio C Young
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Extra resources for Partial differential equations;: An introduction
Sample text
Consider the \I ave equation It is easily seen that the functions u,,(x, y) = (x - y)", 11 = I, 2, ... satisfy the equation for each 11. ::; N. , for k = I, 2, ... i (2k - k=1 •)2k- l I)! \-;·-plane containing the origin. Hence. by the extended principle of superposition, the preceding series is also a solution of the wave equation. As a matter of fact, we notice that the series converges to the function u(x, y) = sin(x - y), which is easily seen to satisfy the wave equation. There is a variation of the principle of superposition concerning solution of a homogeneous differential equation that depends on a parameter.
When x 2 + 2y 2 = 4. + 2xy11 = 0, 11 = c" sin(x + I), when y 2 = 2x + 2 20. r11r 21. ) 11, xu,. 22. rn,. 23. Show that no solution exists for the differential equation of Problelll 22, which assullles the prescribed value ef>(_y) on the circle x 2 + y 2 = a 2 , unless cf> is of the 2 form qJ(x) = kc-x • where k is a constant. If cf> has the indicated form. show that the problem has infinitely many solutions. I.
1. In Problems 7 through 13, find the general solution of the given equation. 7. 9. + + Xllx Jllx Xllx - xy11,. - 2x11 x + 2 2 )'lly - xu = 2e" + x u, - xyu,. + 2y11 2 10. y. xu,. = 2 8. l. Jllr = = 2 x y. 0. 11. y ux - 12. (x 13. xyux - 14. Let Lu = Au, + Bur + Cu= + Du = 0 be a linear equation in three variables x, J', ::, where A, B, C, Dare constants. Introduce the new variables + y)(u, - 11r) + = 11 = y . «YJ. x 2 uy - yu = xy. ~ = a 1x + h1Y + I/ = a1 x + h2Y + ( = G3X + h3y + CI;: C2Z C3Z where the deterlllinant of the coefficients a;.