By Murray R. Spiegel

This quantity prepares readers to translate mathematically utilized difficulties into the language of differential equations, resolve these differential equations topic to given stipulations coming up within the difficulties, and interpret the ideas acquired.

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**Example text**

B EXERCISES Solve each of the following: dy 3y^ cot x + sin x cos x x dy ' dx 3. (3x2 4. (2x ly +J + + ' 3x^y)dx 2xy^)dx + -\- {x^y dx x^y + y^ xdy =0. + 2y + 3f)dy = 0. C EXERCISES 1. , 2. ^^^^^ a function of the product xy, then an integrating factor = xy. Use the u such that /dN 1 xM is method of Exercise (/ + xy 1 to solve + l)dx ^ (x2 + xy + l)dy = is ^J^^")'*" where Sec. 3. First-Order and Simple Higher-Order Equations Ill Solve (ly^ + 4. + 4x^y)dx ing factor of the form In Exercise 2 of the (4xy A x'^y'', + 3:^)dy = 0, 43 given that there exists an integrat- where p and q are constants.

YXO) x)j{,y - x). = 0. >'(-2)=3. : Sec. V Differential Equations in General (d) y = (e) xy- Ci - CoX = Ax - + = 0. x{y'f -f y'" C3X-. B. 2yy' + 23 = xyy" 0. \y'0) Find a 3. differential = 2. equation for each of the following families of curves in the xy plane with center at the origin and any radius, (b) All parabolas (a) All circles through the origin with x axis as common ^ = (d) All ellipses with center at the origin and tangent to the y axis, axes on the coordinate axes. jc axis, (c) All circles with centers on and B EXERCISES Find a differential equation of third order having y = ax sin x -r bx cos x, where a and b are arbitrary constants, as a solution.

Function of y alone. ju is a. may be In this case, (8) =f(x), then I J^'^''^^ state this result as follows: e^^^ ' is an integrating factor. dy — — = — (dN — dM\\dy M\ dx dy du \ -^ or // dy By [say written dN dM ,du — +M—= — ju x alone Cf,^, Theorem. I = /z We may omitting the constant of integration. a function of (9) is so or \f{y)dx Ch. 2 [x ^ ' and we may prove the If The following -— is dM\ (dN 1 Theorem. — I I r... then e^^^"'^ = g(y), mnemonic scheme a an integrating is factor. summarize procedure.

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