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nd Order Linear Ordinary Differential Equations - Inside Mines

12nd Order Linear Ordinary Differential Equations Solutions for Equations of the following general form: dydxaxdydxaxy hx2212++=()()() Reduction of Order If terms are missing from the general second - Order Differential equation, it is sometimes possible to reduce the equation to a first- Order Ordinary Differential equation. second - Order Differential Equations can be solved by reduction of Order for two cases. Dependent Variable (y) is Missing dydxaxdydxhx221+=()() The procedures is to define a new variable p as: dydxp= which can be differentiated again with respect to x to give: dydxdpdx22= These are substituted into the Differential equation to give: dpdxaxp hx+=1()() which can then be solved by integrating factors to give (see handout on solution methods for 1st Order Differential Equations ): ()()()111() ()where exp =+ = phxF xdx IFxFxa xdx The solution y is found by substituting for p = dy / dx and integrating again with respect to x.

1 2nd Order Linear Ordinary Differential Equations Solutions for equations of the following general form: dy dx ax dy dx axy hx 2 2 ++ =12() () Reduction of Order If terms are missing from the general second-order differential equation, it is sometimes possible

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Transcription of nd Order Linear Ordinary Differential Equations - Inside Mines

1 12nd Order Linear Ordinary Differential Equations Solutions for Equations of the following general form: dydxaxdydxaxy hx2212++=()()() Reduction of Order If terms are missing from the general second - Order Differential equation, it is sometimes possible to reduce the equation to a first- Order Ordinary Differential equation. second - Order Differential Equations can be solved by reduction of Order for two cases. Dependent Variable (y) is Missing dydxaxdydxhx221+=()() The procedures is to define a new variable p as: dydxp= which can be differentiated again with respect to x to give: dydxdpdx22= These are substituted into the Differential equation to give: dpdxaxp hx+=1()() which can then be solved by integrating factors to give (see handout on solution methods for 1st Order Differential Equations ): ()()()111() ()where exp =+ = phxF xdx IFxFxa xdx The solution y is found by substituting for p = dy / dx and integrating again with respect to x.

2 2()()()1211() ()where exp =++ = yhxF xdx Idx IFxFxa xdx where I1 and I2 are constants of integration. (Note that throughout this document constants of integration will be indicated by this notation.) Independent Variable (x) is Missing dydxadydxay22120++= The procedures is to define a new variable p as: dydxp= which can be differentiated again with respect to x to give: dydxdpdx22= but remember that p can be written as a function of y which is a function of x. That is p = f(y(x)). This can be differentiated by chain rule to give (remembering that p = dy / dx): dpdxdpdydydxdpdyp== These relationships for p are substituted into the Differential equation to give: pdpdyap a y++=120 The 2nd Order Differential equation of y with respect to x has now been converted into a 1st Order Differential equation of p with respect to y.

3 This equation is nonlinear (because p multiplies dp / dx) and can only be solved analytically if it is possible to separate and integrate. Once p is determined as a function of y ( , p = f(y)), then y can be found by integrating f(y) with respect to x to give: dyfyxI()z=+2 The first constant of integration will be contained in the function f(y). 3 Variation of Parameters This method can be used anytime you already know one solution, yx1(), to the homogeneous form of the general Differential equation given below. dydxaxdydxaxy hx2212++=()()() The complete solution is found by substituting yuxyx=() ()1 into the above Differential equation. The differentials of y are as follows: = + yuyuy11 = + + yuy uyuy1112 which when substituted into the Differential equation gives: + ++ + +=uy u y a xy uy a xy a xyhx1111111212()() ()()bgbg Since y1 is a solution to the homogeneous form of the Differential equation shown at the top of the page, + +=yaxyaxy11 12 10( )( ), and the above equation reduces to give the following Differential equation in u: ()11112() () ++ =uy u y a xyhx The function u can be found from this Differential equation by reducing Order and then solving by integrating factors.

4 The complete solution y can be found by multiplying u by y1 to give the general solution: yuy yyFhxyFdxIyFdxI y==+LNMMOQPP+zz11121212111 () where 11 exp when 0 Fadxa = 1 1 when 0Fa== From which you can identify the second solution and the particular solution as follows: yydxyF2112=z yyyFh x y Fdx dxp=LNMOQPzz11211 () 4 Constant Coefficient Ordinary Differential Equations adydxbdydxcy220++= where a, b and c are constants The form of the Differential equation suggests solutions of yerx=. (At this point this form is deduced by understanding the properties of differentiating erx. Later, we will develop a general approach for determining that this is the form of the solution.) =yrerx =yrerx2 ()arbrc erx20++ = Characteristic Equation: rbb aca= 242 Case 1: Real & Unequal Roots (bac240 >) (a) If rr12 and rr12 , then yIe Ierxr x=+1212 (b) If rrr12= = , then yIe Ierxrx=+ 12 or yA rx Brx=+sinh( )cosh( ) where IAB12=+ and IBA22= Case 2: Complex Roots (bac240 <) r= i , complex conjugate 5where = ba2 and = 422ac ba ye IeIexxx=+ 12i-iej But exxxii=+cossin yeI xixI xixx=++ 12cos()sin()cos()sin()ch Let IAB11 1=+i and IAB22 2=+i and AA A=+12 and BB B= 21 ye Ax Bxx=+ cos()sin()bg Case 3: Real and Equal Roots (bac240 =) rr rba122=== The characteristic equation gives only one solution, yerx1= The second solution can be found by variation of parameters to give.

5 YydxyF2112=z where expexpFbadxbxa=LNMOQP=FHGIKJz yedxeeedxxerxbx a bx arxrx2=== zz yIe Ixerxrx=+12 6 Equidimensional Ordinary Differential Equation dydxaxdydxbxy2220++= where a and b are constants The form of the Differential equation suggests solutions of yxr= (At this point this form is deduced by understanding the properties of differentiating xr. Later, we will develop a general approach for determining that this is the form of the solution.) = yrxr1 = yrr xr()12 rrar bxr() + + = 102bg Characteristic Equation: rarb210+ +=() raab= 11422() Case 1: Real & Unequal Roots ()1402 >ab yIx Ixrr=+1212 Case 2: Complex Roots ()1402 <ab r= i where = 12a and = 41222ba() yIxx IxxIxeIxexx=+ = + 12 12 iiiilnln But exxxii=+cossin Let IAB11 1=+i and IAB22 2=+i and AA A=+12 and BB B= 21 yAxx Bxx=+ cos( ln )sin( ln ) 7 Case 3: Real and Equal Roots ()1402 =ab rr ra1212=== The characteristic equation gives only one solution, yxr1= The second solution can be found by variation of parameters to give.

6 YydxyF2112=z where expexp lnFaxdxaxxa=LNMOQP==zbg yxdxxxxdxxxxdxxxxraaraarr22121===== zzz()/()ln yIx Ix xrr=+12ln 8 Special Equations Several second - Order Ordinary Differential Equations arise so often that they have been given names. Some of these are listed below. Harmonic Equation The following Differential equation commonly arises for problems written in a rectangular coordinate system. dydxby2220+= where b2 is not a function of x or y This Differential equation is a constant coefficient equation with the solution: yI bx Ibx=+12sin()cos() Modified Harmonic Equation Like the harmonic equation, this equation commonly arises for problems written in a rectangular coordinate system. dydxby2220 = where b2 is not a function of x or y This Differential equation is a constant coefficient equation with the solution: yIbx Ibx=+12sinh()cosh() This equation can also be written in terms of the exponential as: yIbx Ibx=+ 12exp()exp() As a general rule, it is usually convenient to use the sinh/cosh form of the solution for problems with finite boundaries and to use the exponential form of the solution for problems with one or more infinite boundaries.

7 Bessel's Equation The following Differential equation commonly arises for problems written in a cylindrical coordinate system. xdydxxdydxbx p y2222220++ =ej 9where b2 and p2 are constants. This Differential equation has the solution yAJbx BJ bxpp=+ ()() where A and B are constants of integration, and Jp is the Bessel function of the first kind and Order p. If p is an integer or if p = 0, then the Differential equation is: xdydxxdydxbx n y2222220++ =ej where n is an integer or zero. The solution to this equation is: yAJbx BYbxnn=+()() where Yn is the Bessel function of the second kind and Order n. The Bessel functions of the first and second kind are similar to the sine and cosine functions ( , solutions to the harmonic equation). In particular, like the sine and cosine functions, Bessel functions of the first and second kind are periodic for real arguments.

8 Modified Bessel's Equation Like Bessel's equation, this equation commonly arises for problems written in cylindrical a coordinate system. xdydxxdydxbx p y2222220+ + =ej where b2 and p2 are constants. This Differential equation has the solution yAIbx BI bxpp=+ ()() where A and B are constants of integration, and Ip is the modified Bessel function of the first kind and Order p. If p is an integer or if p = 0, then the Differential equation is: xdydxxdydxbx n y2222220+ + =ej where n is an integer or zero. The solution to this equation is: yAIbx BKbxnn=+()() where Kn is the modified Bessel function of the second kind and Order n. Modified Bessel functions of the first and second kind are similar to the hyperbolic sine and hyperbolic cosine functions ( , solutions to the modified harmonic equation).

9 Most importantly, modified Bessel functions of the first and second kind are not periodic functions. 10 Special Functions Error Function A number of physical problems of interest to chemical engineers will produce Equations in which dydxex= 2 which has the solution yxdxI= +zexp2ej where I is a constant of integration. To clarify the exact operation that is intended by the above equation, it is better to write the solution as: ysdsIx= +zexp20ej The integral of the exp(-s2) must be determined numerically, except when x is infinity, in which case exp = zsds202ej Because problems with this type of solution arise frequently, it was convenient to define a function that represents this integral. The name of this function is the Error Function and it is defined as: erf xsdsx()exp= z220 ej By defining it in this way, erf(x) = 0 when x = 0 and erf(x) = 1 when x.

10 Using the error function, the solution to the Differential equation at the top of this page is: yerfxI=+ 2() Complementary Error Function The complementary error function erfc(x) is defined as: erfc xerf x()()= 1


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