engineering math

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engineering math

PartiOrdinary Differential EquationsI Introduction toDifferential EquationsEXERCISES 1.1Definitions and Terminology1Second order: linear2Thin! order;

engineering math nonlinear because of (dy/dx)43Fourth order; linear4Second order: nonlinear because of cos(r 4- u)5Second order: nonlinear because of (dy/dx)~ or ỵ 1

4- {dyidx)26Second order; nonlinear because of li17Third order; linear8Second order: nonlinear because of X‘9Writing the differential equation in the engineering math

form x{dy/dx) 4- J/2 = 1, we -sec that it is nonlinear in y l>ecause of J/2. However, writing it in the form (»/2 - ỉ)(dx/dy) + X = 0. we see that it

engineering math

is linear in X.10Writing the differential equation in the form u(dv/d«) 4- (1 4- ti)v = ueu we see that it is linear in V. However, writing it in the

PartiOrdinary Differential EquationsI Introduction toDifferential EquationsEXERCISES 1.1Definitions and Terminology1Second order: linear2Thin! order;

engineering math we obtain dyịdt = ide”8®1, so that4- 20?/ = 24e“ao< + 20 (I - |e-20') = 24.13From y = earcos2x we obtain y1 = 3e32cos2x - 2c3rsin2x and y" = 5c3rcos2

x - 12c3* sin 2x, so that .v" - 6/ 4- 13.V = 0.14Hom y = - C08X ln(secx + tan x) we obtain y = -1 4- sin X hi(secx 4- tanx) and y" = tanx + cosxln(sec engineering math

x 4- tanx). Then y" 4- y = tanx.15The domain of the function, found by solving X 4- 2 > 0, is [—2,oo). From /=14- 2(x 4- 2)-I?’2 we have(y - x)ự = (y-

engineering math

x)[l + (2(x 4- 2)-1/3|= y - X 4- 2(y - x)(x 4- 2)~1/3= y - X 4- 2[x 4- 4(x 4- 2)l/2 - x](x 4- 2)-i/2= y - X 4- 8(x 4- 2)l/2(x 4- 2)-1/2 = y - X 4- 8.

PartiOrdinary Differential EquationsI Introduction toDifferential EquationsEXERCISES 1.1Definitions and Terminology1Second order: linear2Thin! order;

engineering math .16Since tanx is not defined for X = rr/2 + niĩ, n an integer, the domain of y = 5tan5x is {x I 5of / x/2 + nr} or {x I X Ỷ r/10 + nr/5}. FYotii ỊỈ =

25sec25ar we haveỳ = 25( 1 + tan2 5z) = 25 + 25 tan2 5x = 25 + y2.All interval of definition for the solution of the differential equation is (-ít/IO. engineering math

ít/IO). Another interval is (jr/10,3ir/10), and so on.17The domain of the function is {x I 4 - X2 / 0} or {x I X / -2 or X / 2}. Roll! y' = 2x/(4 - X2

engineering math

)2 we have= 2xy.All interval of definition for the solution of the differential equation is (-2.2). Other intervals are (-00,-2) and (2,oc).18The func

PartiOrdinary Differential EquationsI Introduction toDifferential EquationsEXERCISES 1.1Definitions and Terminology1Second order: linear2Thin! order;

engineering math 2(- cosx) we have

PartiOrdinary Differential EquationsI Introduction toDifferential EquationsEXERCISES 1.1Definitions and Terminology1Second order: linear2Thin! order;