Math 2Z03 - Tutorial # 6. Oct. 26th, 27th, 28th, 2015
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1 Math 2Z03 - Tutorial # 6 Oct. 26th, 27th, 28th, 2015
2 Tutorial Info: Tutorial Website: dedieula/2z03.html Office Hours: Mondays 3pm - 5pm (in the Math Help Centre)
3 Tutorial #6: 3.4 Undetermined Coefficients
4 Tutorial #6: 3.4 Undetermined Coefficients 3.5 Variation of Parameters
5 3.4 Undetermined Coefficients Undetermined Coefficients: a method of solution for finding a particular solution to a linear differential equation with constant coefficients a n y (n) + + a 1 y + a 0 y = g(x), where g(x) is a polynomial, exponential e ax, sine, cosine, or some sum/product of these function.
6 3.4 Undetermined Coefficients Method of Solution:
7 3.4 Undetermined Coefficients Method of Solution: 1. Find the general solution, y c, to the associated homogeneous equation a n y (n) + + a 1 y + a 0 y = g(x).
8 3.4 Undetermined Coefficients Method of Solution: 1. Find the general solution, y c, to the associated homogeneous equation a n y (n) + + a 1 y + a 0 y = g(x). 2. Choose a trial particular solution, y pi for each term in g(x).
9 3.4 Undetermined Coefficients Method of Solution: 1. Find the general solution, y c, to the associated homogeneous equation a n y (n) + + a 1 y + a 0 y = g(x). 2. Choose a trial particular solution, y pi for each term in g(x). 3. If any y pi contains terms that duplicate terms in y c, then multiply y pi by x n, where n is the smallest possible integer that eliminate the duplication.
10 3.4 Undetermined Coefficients Method of Solution: 1. Find the general solution, y c, to the associated homogeneous equation a n y (n) + + a 1 y + a 0 y = g(x). 2. Choose a trial particular solution, y pi for each term in g(x). 3. If any y pi contains terms that duplicate terms in y c, then multiply y pi by x n, where n is the smallest possible integer that eliminate the duplication. 4. Plug the sum of the terms found in Step 3, y p, into the original DE and solve for the undetermined coefficients.
11 3.4 Undetermined Coefficients Method of Solution: 1. Find the general solution, y c, to the associated homogeneous equation a n y (n) + + a 1 y + a 0 y = g(x). 2. Choose a trial particular solution, y pi for each term in g(x). 3. If any y pi contains terms that duplicate terms in y c, then multiply y pi by x n, where n is the smallest possible integer that eliminate the duplication. 4. Plug the sum of the terms found in Step 3, y p, into the original DE and solve for the undetermined coefficients. 5. The general solution is y = y c + y p.
12 3.4 Undetermined Coefficients Trial Particular Solutions:
13 3.4 Undetermined Coefficients Trial Particular Solutions: A polynomial of degree k corresponds to a trial solution a 0 + a 1 x + + a k x k.
14 3.4 Undetermined Coefficients Trial Particular Solutions: A polynomial of degree k corresponds to a trial solution a 0 + a 1 x + + a k x k. e mx corresponds to trial solution Ae mx.
15 3.4 Undetermined Coefficients Trial Particular Solutions: A polynomial of degree k corresponds to a trial solution a 0 + a 1 x + + a k x k. e mx corresponds to trial solution Ae mx. sin(mx) and cos(mx) both correspond to the trial solution Acos(mx) + Bsin(mx).
16 3.4 Undetermined Coefficients Trial Particular Solutions: A polynomial of degree k corresponds to a trial solution a 0 + a 1 x + + a k x k. e mx corresponds to trial solution Ae mx. sin(mx) and cos(mx) both correspond to the trial solution Acos(mx) + Bsin(mx). If you have a product of the above, then take the corresponding product of trial solutions. e.g. xe 3x cos(4x) corresponds to trial solution (a 0 + a 1 x)(ae 3x )(Bcos(4x) + Csin(4x)) = (c 0 + c 1 x)(e 3x )(Bcos(4x) + Csin(4x)).
17 3.4 Undetermined Coefficients 1. For each of the following, the general solution of the associated homogenous equation is given. What form will its particular solution have?
18 3.4 Undetermined Coefficients 1. For each of the following, the general solution of the associated homogenous equation is given. What form will its particular solution have? a) y + 3y = 48x 2 e 3x ; y c = c 1 cos( 3x) + c 2 sin( 3x)
19 3.4 Undetermined Coefficients 1. For each of the following, the general solution of the associated homogenous equation is given. What form will its particular solution have? a) y + 3y = 48x 2 e 3x ; y c = c 1 cos( 3x) + c 2 sin( 3x) b) y 6y = 3 cosx; y c = c 1 + c 2 x + c 3 e 6x
20 3.4 Undetermined Coefficients 1. For each of the following, the general solution of the associated homogenous equation is given. What form will its particular solution have? a) y + 3y = 48x 2 e 3x ; y c = c 1 cos( 3x) + c 2 sin( 3x) b) y 6y = 3 cosx; y c = c 1 + c 2 x + c 3 e 6x c) y y y = 3 + e x 2 ; y c = c 1 e x 2 + c 2 xe x 2
21 3.4 Undetermined Coefficients 1. For each of the following, the general solution of the associated homogenous equation is given. What form will its particular solution have? a) y + 3y = 48x 2 e 3x ; y c = c 1 cos( 3x) + c 2 sin( 3x) b) y 6y = 3 cosx; y c = c 1 + c 2 x + c 3 e 6x c) y y y = 3 + e x 2 ; y c = c 1 e x 2 + c 2 xe x 2 2. Find the general solution of the differential equations in #1.
22 3.5 Variation of Parameters Variation of Parameters: a method of solution for finding a particular solution to a linear differential equation a n (x)y (n) + + a 1 (x)y + a 0 (x)y = g(x). To use it, we must already have a general solution for the corresponding homogeneous equation a n (x)y (n) + + a 1 (x)y + a 0 (x)y = 0
23 3.5 Variation of Parameters Variation of Parameters: a method of solution for finding a particular solution to a linear differential equation a n (x)y (n) + + a 1 (x)y + a 0 (x)y = g(x). To use it, we must already have a general solution for the corresponding homogeneous equation a n (x)y (n) + + a 1 (x)y + a 0 (x)y = 0 Advantages Over Method of Undetermined Coefficients:
24 3.5 Variation of Parameters Variation of Parameters: a method of solution for finding a particular solution to a linear differential equation a n (x)y (n) + + a 1 (x)y + a 0 (x)y = g(x). To use it, we must already have a general solution for the corresponding homogeneous equation a n (x)y (n) + + a 1 (x)y + a 0 (x)y = 0 Advantages Over Method of Undetermined Coefficients: For Undetermined Coefficients, we require that g(x) is a sum/product of polynomials, e ax s, sines, and cosines. For Variation of Parameters, we require no such restrictions on g(x).
25 3.5 Variation of Parameters Variation of Parameters: a method of solution for finding a particular solution to a linear differential equation a n (x)y (n) + + a 1 (x)y + a 0 (x)y = g(x). To use it, we must already have a general solution for the corresponding homogeneous equation a n (x)y (n) + + a 1 (x)y + a 0 (x)y = 0 Advantages Over Method of Undetermined Coefficients: For Undetermined Coefficients, we require that g(x) is a sum/product of polynomials, e ax s, sines, and cosines. For Variation of Parameters, we require no such restrictions on g(x). Undetermined Coefficients requires that the linear DE has constant coefficients, whereas Variation of Parameters does not.
26 3.5 Variation of Parameters Method of Solution: I write the method of solution for second-order linear DE s, but this method naturally generalizes for higher order DE s.
27 3.5 Variation of Parameters Method of Solution: I write the method of solution for second-order linear DE s, but this method naturally generalizes for higher order DE s. 1. Find the general solution, y c = c 1 y 1 + c 2 y 2, for corresponding homogeneous equation a 2 (x)y + a 1 (x)y + a 0 (x)y = 0.
28 3.5 Variation of Parameters Method of Solution: I write the method of solution for second-order linear DE s, but this method naturally generalizes for higher order DE s. 1. Find the general solution, y c = c 1 y 1 + c 2 y 2, for corresponding homogeneous equation a 2 (x)y + a 1 (x)y + a 0 (x)y = Compute Wronskian W(y 1 (x),y 2 (x)).
29 3.5 Variation of Parameters Method of Solution: I write the method of solution for second-order linear DE s, but this method naturally generalizes for higher order DE s. 1. Find the general solution, y c = c 1 y 1 + c 2 y 2, for corresponding homogeneous equation a 2 (x)y + a 1 (x)y + a 0 (x)y = Compute Wronskian W(y 1 (x),y 2 (x)). 3. Put equation in standard form y + P(x)y + Q(x)y = f (x).
30 3.5 Variation of Parameters Method of Solution: I write the method of solution for second-order linear DE s, but this method naturally generalizes for higher order DE s. 1. Find the general solution, y c = c 1 y 1 + c 2 y 2, for corresponding homogeneous equation a 2 (x)y + a 1 (x)y + a 0 (x)y = Compute Wronskian W(y 1 (x),y 2 (x)). 3. Put equation in standard form y + P(x)y + Q(x)y = f (x). 4. Compute W 1 = 0 y 2 f (x) y, W 2 = y y 1 f (x).
31 3.5 Variation of Parameters Method of Solution: I write the method of solution for second-order linear DE s, but this method naturally generalizes for higher order DE s. 1. Find the general solution, y c = c 1 y 1 + c 2 y 2, for corresponding homogeneous equation a 2 (x)y + a 1 (x)y + a 0 (x)y = Compute Wronskian W(y 1 (x),y 2 (x)). 3. Put equation in standard form y + P(x)y + Q(x)y = f (x). 4. Compute W 1 = 0 y 2 f (x) y, W 2 = y y 1 f (x). 5. Find u 1 := W 1 W dx and u 2 := W 2 W dx.
32 3.5 Variation of Parameters Method of Solution: I write the method of solution for second-order linear DE s, but this method naturally generalizes for higher order DE s. 1. Find the general solution, y c = c 1 y 1 + c 2 y 2, for corresponding homogeneous equation a 2 (x)y + a 1 (x)y + a 0 (x)y = Compute Wronskian W(y 1 (x),y 2 (x)). 3. Put equation in standard form y + P(x)y + Q(x)y = f (x). 4. Compute W 1 = 0 y 2 f (x) y, W 2 = y y 1 f (x). 5. Find u 1 := W 1 W dx and u 2 := W 2 W dx. 6. A particular solution is y p = u 1 y 1 + u 2 y 2, and the general solution is y = y c + y p.
33 3.5 Variation of Parameters 3. Find the general solution of x 2 y + xy + (x )y = x 3 2, where y 1 = x 1 2 cosx and y 2 = x 1 2 sinx and linearly independent solutions of the associated homogeneous DE on (0, ).
34 3.5 Variation of Parameters 3. Find the general solution of x 2 y + xy + (x )y = x 3 2, where y 1 = x 1 2 cosx and y 2 = x 1 2 sinx and linearly independent solutions of the associated homogeneous DE on (0, ). 4. Solve y + y = tanx.
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