M(x, y) dx+ N(x, y)dy=0 (1) In general, Eq. (1) is not exact. Occasionally, it is possible to transform (1) into an exact differential equation by a judicious multiplication. A function /(x, y) is an integrating factor for (1) if the equation 1(x, y)[M(x, y) dx + N(x, y)dy] =0 (5.7) is exact. A solution to (1) is obtained by solving the exact differential equation defined by (2) Some of the more common integrating factors are displayed in Table 5-1 and the conditions that follow: If Integrating Factors ᎥᏝ 1 M ƏN N dy əx ƏM ƏN ax M dy g(x), a function of x alone, then 1(x, y) = elgond h(y), a function of y alone, then If M=yf(xy) and N=xg(xy), then 1(x, y) = eth(y)dy 1(x, y)=- xM-yN
Problems In Problems 5.41 through 5.55, find an appropriate integrating factor for each differential equation and solve 5.41. (y+1)d=xdy=0 5.42. y dx + (1-x) dy=0 5.43 (x+y+y) dx-x dy=0 5.44. (y+x³y³) dx + x dy=0
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