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 coming up within the difficulties, and interpret the options got.

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Y (c) x^ (e) (f) (g) (i) (j) 2. = 3;c^ + ce~'^^\ c. (b) y = X + c sin x: c. — ay^ = a. (d) ylnx = bx: b. y = Ci sin 4x + C2 cos 4x + x: c^, Cg. / = oite-^ + i3e-* + 2 sin 3/: a, (h) y = ax^ + ftx^ + ex: x'^ + y^ — ex = 0: c. y = /4e-3* + 5^2^ + Ce*^: ^, 5, C. (a) 1 : /?. r = Show + a In 60 + c: a, b, c. Column I is a general solution of the differential Obtain particular solutions satisfying conditions in that each relation in equation of Column Column 11. III. (b) (c) y = A y x^ = y' cos X. c^e^ + — y^ + Ill II I (sl) a, b, c.

To obtain = 2x — \ = m and realizing that for any Une Ix - \ = m have associated lineal elements setting the 1 1 following, (a) y' in (o. m is someUse the method of of slope m. This isoclines (constant slopej. v. F cz are real, single-valued. An 26 Differential F'-'-ations in Ch. General I at all points {x,y,. ) within a region R of 3-dimensional space, then there exists one and only one solution y = g{x) which passes through any = given point of R. -^o) this is equivalent Xo y and continuous y^, y' = Zq for X = Xq.

Y-,v observed that this integration. A. in finding/(v)- on page 30. Y-/(y) (11) First-Order and Simple Higher-Order Equations 32 All we have to do now is show that there exists a function/O') such that (11) second of equations will also satisfy the Ch. , we must show that there is a function /(y) such that dyU To show Mdx+f{y) = N we need only prove this, or -— M dx =N f'(y) \ that \Mdx N-— -I' dvJ dy is (12) a function of y alone. This will indeed be true respect to x of the expression (12) is But zero.