By Murray R. Spiegel

This quantity prepares readers to translate mathematically utilized difficulties into the language of differential equations, clear up these differential equations topic to given stipulations bobbing up within the difficulties, and interpret the suggestions received.

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**Example text**

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. Consider to Mdx + Ndy = Compute ox dy If (1) = If (1) ^ the equation (2), compute (2), is (1) exact and can easily be solved. minus (2), divided by A''; call the result /.

Solve each of the following: ^^^%^~x /— ay o ^-^ (c) (e) /' + =*• (b)x/ + 3j=x2. „ + xy = 2/ + -^ 3/ = dv (d) 1. -^ e-2'; 7(0) = 5. (f) (g)/ = j4l^. 2. The current -f dx y' - 2v -^ X = x2 sin 3x. + y coX x = cos x. ""I- in amperes, in a certain electric circuit satisfies the differential 7, equation dl - +11 = 10e-2« dt where t is the time. If 7 = where t = 0, find 7 as a function of t. B EXERCISES dy 1. The equation — -\- Py = Qy^, where P and Q are functions of x alone and n is a constant, is called BernoulWs differential equation.

Y coX x = cos x. ""I- in amperes, in a certain electric circuit satisfies the differential 7, equation dl - +11 = 10e-2« dt where t is the time. If 7 = where t = 0, find 7 as a function of t. B EXERCISES dy 1. The equation — -\- Py = Qy^, where P and Q are functions of x alone and n is a constant, is called BernoulWs differential equation. « =0or 1. Show how to solve it where rt 7^ 0, 1, none of the methods discussed so far applies. Show, however, that by changing the dependent variable from yXov according to the transformation 2.