Final Exam Review

Question 1

How do you solve a differential equation using integrating factor?

Answer 1

1. Get equation in standard form dy/dx+P(x)y=f(x) 2. Integrate integrating factor P(x) -> e^int(P(x)) dx 3. Multiply both sides by integrating factor result R(x) 4. LHS becomes d/dx[R(x)*y] 5. Integrate both sides and solve for y

Question 2

How do you solve a differential equation using exact equations method?

Answer 2

1. Get equation in standard form M(x,y) dx + N(x,y) dy = 0 2. Integrate M with respect to y and add g(y), this is f(x,y): f(x,y) = int(M(x,y) + g(y) 3. Take the partial of #2 with respect to y to get pf/py = p/py[int(M(x,y)] + g’(y), equate to N(x,y) and solve for g’(y). 4. Integrate the result of #3 with respect to y and solve for g(y) 5. Plug g(y) back into 2, and set entire result = c

Question 3

How do you solve a differential equation using substitution (non-Bernoulli)?

Answer 3

1. Let least complicated variable = v or u times the other variable 2. Find the derivative of v or u (u dx + xdu or v dy + y dv) 3. Substitute these equivalences into original equations 4. Attempt to separate variables 5. Integrate as necessary 6. Switch y/x back in for u or x/y back in for v 7. Solve using algebra

Question 4

How do you solve a differential equation using subsitution (Bernoulli)?

Answer 4

1. Get equation in standard form dy/dx + P(x)y = y^n[f(x)]\ 2. Let u = y^(1-n), and solve this equality for y 3. Find substitution for dy/dx by finding dy/du to get some f(u) times du/dx 4. Plug both substitutions back into equation 5. Solve using integrating factor 6. Plug substitutions into solution to return equation back to dy and y.

Question 5

How do you solve a differential equation using subsitution (Bernoulli)?

Answer 5

1. Get equation in standard form dy/dx + P(x)y = y^n[f(x)]\ 2. Let u = y^(1-n), and solve this equality for y 3. Find substitution for dy/dx by finding dy/du to get some f(u) times du/dx. Make sure for du/dx you plug u=f(y) back in for all values of u. 4. Plug both substitutions back into equation 5. Solve using integrating factor 6. Plug substitutions into solution to return equation back to dy and y.

Question 6

How do you test if a solution is linearly independent?

Answer 6

Use the Wronskian,

Question 7

How do you find the Wronskian?

Answer 7

1. Create an m x n matrix, where m is the order of the highest order derivative and n is the number of functions in the solution set you wish to test linear independence 2. Take the determinant of the matrix. If it is not equal to 0, the solutions are linearly independent.

Question 8

What is a fundamental set of solutions?

Answer 8

Any set (y1, y2 … yn) of n solutions of a homogeneous linear n-th order DE on an interval I is a fundamental set of solutions.

Question 9

What is a general solution?

Answer 9

y=c1y1(x) + c2y2(x) + … + cnyn(x) + yp

Question 10

What is a linear combination?

Answer 10

Any combination of c1y1(x) + c2y2(x) + … + cnyn(x)

Question 11

What is superposition in terms of a homogeneous equation?

Answer 11

You can take any solutions of the differential equation and add them together. Ex: y1, y2, yn are solutions for a DE. You can say the solution is y=y1+y2+yn and it will still work.

Question 12

What is superposition in terms of a nonhomogeneous equation?

Answer 12

You can take the RHS terms of a set of nonhomogeneous equations with the same LHS, add them together, and then add together the particular solution, and the particular solution will be valid for this new additive equation.

Question 13

What is the equation for second solution?

Answer 13

y2=y1(x) * int((e^-int(P(x) dx)/(y1(x)^2 dx))

Question 14

What is the P(x) term in second solution?

Answer 14

The function multiplying the first derivative in the DE. Ex: y’’ + 3xy’+2=0. 3x = P(x).

Question 15

For homogeneous second order and higher linear equations with constant coefficients and distinct real roots for the auxiliary equation am^2+bm+c=0, what is the solution?

Answer 15

y=c1e^m1x + c2e^m2x For higher, it is just the zeroes put into the “m” values for the c1e^mnx terms.

Question 16

For homogeneous second order and higher linear equations with constant coefficients and repeated roots for the auxiliary equation am^2+bm+c=0, what is the solution?

Answer 16

y=c1e^mx + c2xe^mx

Question 17

For homogeneous second order and higher linear equations with constant coefficients and complex roots for the auxiliary equation am^2+bm+c=0, what is the solution?

Answer 17

y=e^alphax * (c1 cos beta x + c2 sin beta x) where the roots are alpha + i(beta) and alpha - i(beta).

Question 18

What is the general solution of a DE in the form y’‘+k^2y = 0?

Answer 18

y=c1 cos kx + c2 sin kx

Question 19

What is the general solution of a DE in the form y’‘-k^2y=0

Answer 19

y=c1e^kx + c2e^-kx

Question 20

What does it mean if a function is analytic at a point a?

Answer 20

The function’s power series representation either converges for all real numbers, or converges within the radius of convergence in relation to the point. If it diverges for all real numbers or for that point, it is not analytic.

Question 21

What is an ordinary point of a DE?

Answer 21

That both coefficients P(x) and Q(x) in y’’ + P(x)y’ + Q(x)y = 0 (obtained by dividing the DE by the leading coefficient in a2(x)y’’ + a1(x)y’ +a0(x)y=0) are analytic at x0

Question 22

What is a singular point of a DE?

Answer 22

A point that is not an ordinary point of the differential equation a2(x)y’‘+a1(x)y’+a0(x)y=0?

Question 23

What is a regular singular point of DE?

Answer 23

A singular point for which the functions p(x)=(x-x0)P(x) and q(x)=(x-x0)^2Q(x) are both analytic at x0 (P(x) and Q(x) are obtained by dividing the entire equation by the coefficient on y’’). For x-x0 to be a singular point, the factor x-x0 can appear at most to the first power in the denominator of P(x) and at most to the second power in the denominator of Q(x).

Question 24

What is the method of Frobenius?

Answer 24

If x=x0 is a regular singular point of the differential equation, then there exists at least one solution of the form shown where the number r is a constant to be determined. The series will converge at least on some interval 0 < x - x0 < R. If r is found to be a number that is a negative integer

Question 25

How do you solve a differential equation using the method of undetermined coefficients and annihilator approach?

Answer 25

1. Determine complementary solution yc by solving the LHS as a homogeneous equation.
2. Determine the remainder of the terms by determining the annihilator function that will eliminate the RHS (if RHS is nonhomogeneous)
3. The remainder of the terms are yp. Determine a function using A,B,C,D,E…Z as coefficients to replace the C1, C2 coefficients of yp.
4. Determine derivative and second derivative of this function as necessary.
5. Plug the derivatives of yp into the original equation for y.
6. Solve for coefficients, and plug the determined coefficients into the general solution for the particular portion.

Question 26

What annihilates 1, x, x^2, x^(n-1)?

Answer 26

D^n

Question 27

What annihilates e^(alpha)x, xe^(alpha)x, x^2e^(alpha)x, x^(n-1)e^(alpha)x?

Answer 27

(D - (alpha)^n

Question 28

What annihilates e^(alpha)x sin/cos (beta)x, xe^(alpha)x sin/cos (beta)x, etc?

Answer 28

[D^2-2(alpha)D+((alpha)^2 + (beta)^2))]^n. If (alpha) = 0 and n=1, (D^2 + (beta)^2) annihilates sin/cos(beta)x

Question 29

How do you solve a differential equation using variation of parameters?

Answer 29

1. Get the equation into standard form y’’ + P(x)y’ + Q(x)y=f(x)
2. Determine the complementary solution using the homogeneous part from the LHS
3. Y1 = The first non-coefficient function from the complementary solution, Y2 = the second non-coefficient function from the complementary solution.
4. Find the Wronskian with Y1 in the first column and Y2 in the second
5. Yp = -y1 * int[(y2*f(x))/W] + y2 * int[(y1*f(x)/W]
6. Combine complementary solution and particular solution