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# Class Note for MATH 3321 at UH

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Date Created: 02/06/15
22 Separable Equations A rst order differential equation 3 m y is a separable equation if the function f can be expressed as the product of a function of z and a function of y That is the equation is separable if the function f has the form Wm 24 1990 719 where p and h are continuous functions The solution method for separable equations is based on writing the equation as qyy W 1 where qty 17104 Of course in dividing the equation by My we have to assume that My 7 0 Any numbers r such that hr 0 may result in singular solutions of the form y r If we write 3 as dydm and interpret this symbol as differential 3 divided by differential m then a separable equation can be written in differential form as This is the motivation for the term separable the variables are separated Solution Method for Separable Equations Step 1 Identify Can you write the equation in the form If yes do so In expanded form equation 1 is Step 2 Integrate this equation with respect to z y m dz dz C C an arbitrary constant which can be written qltygtdypltxgtdz0 by setting y and dy y m dm Now if P is an antiderivative for p and if Q is an antiderivative for q then this equation is equivalent to Qty PW C 2 29 INTEGRAL CURVES Equation 2 is a oneparameter family of curves called the integral curves of equation In general the integral curves de ne y implicitly as a function of z These curves are solutions of 1 since by implicit differentiation le wmw ici qyy 2990 Example 1 The differential equation 1 y W 24 0 y 739 is separable since fm y Writing the equation in the form 1 yy7m or ydy7mdm ydy7xdz0 y27m20 or m2y20 and integrating we get which after multiplying by 2 gives m2y22C or z2y2C Since C is an arbitrary constant 20 is arbitrary and so we ll just call it C again This manipulation of arbitrary constants is standard in differential equations courses you ll have to become accustomed to it The set of integral curves is the family of circles centered at the origin Note that for each positive value of C the resulting equation defines y implicitly as a function of m I Remark we may or may not be able to solve the implicit relation 2 for y This is in contrast to linear differential equations where the solutions y are given explicitly as a function of m See equation 2 in Section 21 When we can solve 2 for y we will As we shall illustrate below the set of integral curves of a separable equation may not represent the set of all solutions of the equation and so it is not technically correct to use the term general solution77 as we did with linear equations However for our purposes here this is a minor point and so we shall also call 2 the general solution of As noted above if hr 0 then y r may be a singular solution of the equation we will have to check for singular solutions 30 Example 2 Show that the differential equation myiy 1 y y1 w is separable Then 1 Find the general solution and any singular solutions 2 Find a solution which satis es the initial condition y2 1 SOL UTION Here zyiy 2495 1 y 7 717 may y1 y1 90 y1 Thus7 f can be expressed as the product of a function of z and a function of y so the equation is separable Writing the equation in the form 17 we have y1 729671 247W 1 17 y z71 y the variables are separated lntegrating with respect to x we get 1gt dyz71dz0 ylnlylm27x0 or and is the general solution Again we have y de ned implicitly as a function of z Note that y 0 is a solution of the differential equation verify this7 but this function is not included in the general solution ln 0 does not exist Thus7 y 0 is a singular solution of the equation To nd a solution that satis es the initial condition7 set x 27 y 1 in the general solution 1ln 1 2272C which implies 01 A particular solution that satis es the initial condition is y 1 ln 2 7 z 1 I Example 3 Show that the differential equation 2 eyim 31 is separable and find the general solution SOLUTION Since ey m e mey we have fzy zey 57159 Thus 1 can be expressed as the product of a function of z and a function of y the equation is separable We have 3 zey m maimey To write the equation in the form 1 divide by 5y ie multiply by 57y to get 7y m e yxe Note Since e y 7 0 for all y there will be no singular solutions lntegrating this equation with respect to x we get e ydy meimdz0 7572 ime m 7 57m C integration by parts 67y me m 67x C the general solution Again the general solution gives y implicitly as a function of m However in this case we can solve for y by applying the natural log function 7y ln 67w 57m C y ilnmeimeim0 I Example 4 As you can check the differential equation y zy2z is both linear and separable so it can be solved using either method We ll solve it as a separable equation You should also solve it as a linear equation and compare the two approaches Rewriting the equation in the form 1 we have 1 my 95 WED lntegrating this equation we get 1 d d c 7 13 13 y 2 y ln ly 2 m2 C general solution 32 We can solve the latter equation for y as follows lnly2l m2C ly 2 5122C eC emzZ Key2 K so an arbitrary constant By allowing K to take on both positive and negative values7 we can write the general solution as y2Kem22 or yKem2272 If we set K 0 we get y 72 so this solution is included in the general solution y 72 is not a singular solution Exercises 22 Find the general solution and any singular solutions If possible7 express your general solution in the form y 1 y mylZ 2 y y sin2m 3 3 y 3x21 yz 4 ylizyz x 5 dig7 sinzy 39dmilimz39 2 6 y y 1 WM 7 y mem 8 y my27x7y21 g dy71y2 39dm71m239 dy y 10l i7 nmdm m 2 1 11 ylnmy y z 1239diy7sin139 dz mzycosy Find a solution of the initial value problem 33 13 9021 y 7 my 2471 1 14 y 652m y y0 0 15 my 7 y 2m2y7 y1 1 dy em y 16 7 1 0 dm 1 em7 ylt 17 7amp1 7 9 371 y i y 1 7 y i 1 7 y2 18 y m172 y0 0 Find the general solution These equations are a mixture of linear7 separable and Bernoulli equations 19 m1 yz y1 m2y 0 20 my y 25 21 my y7secz0 22 my 1 yy m 7 my 23 7 2m 7 2mg 24 y 2mg 2x 25 3m2 1y 7 2mg 6x 1 26 m17 y y1 355 0 27 my 1 y yz ln z 28 y 7379 z4y13 z HOMOGENEOUS EQUATIONS A rst order differential equation 3 fzy is homogeneous if Am7 Ag m y for all 7 O The change of variable de ned by y um dydm vzdvdz transforms a homogeneous equation into a separable equation dy do a 7 fzy becomes vz 7 fzvm 7 f1v which is a separable equation in z and U Show that each of the following differential equations is homogeneous and nd the general solution of the equation 2 2 29 d7y 3 1y dm 2mg 34 30 31 32 y 33 34 Ly L dm x dy 7 mzeym y2 dm 7 my 39 4 2294 my 39 y asinltagt I I yW m 35

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