Final Exam Review Flashcards
How do you solve a differential equation using integrating factor?
- 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
How do you solve a differential equation using exact equations method?
- 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
How do you solve a differential equation using substitution (non-Bernoulli)?
- 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
How do you solve a differential equation using subsitution (Bernoulli)?
- 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.
How do you solve a differential equation using subsitution (Bernoulli)?
- 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.
How do you test if a solution is linearly independent?
Use the Wronskian,
How do you find the Wronskian?
- 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.
What is a fundamental set of solutions?
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.
What is a general solution?
y=c1y1(x) + c2y2(x) + … + cnyn(x) + yp
What is a linear combination?
Any combination of c1y1(x) + c2y2(x) + … + cnyn(x)
What is superposition in terms of a homogeneous equation?
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.
What is superposition in terms of a nonhomogeneous equation?
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.
What is the equation for second solution?
y2=y1(x) * int((e^-int(P(x) dx)/(y1(x)^2 dx))
What is the P(x) term in second solution?
The function multiplying the first derivative in the DE. Ex: y’’ + 3xy’+2=0. 3x = P(x).
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?
y=c1e^m1x + c2e^m2x For higher, it is just the zeroes put into the “m” values for the c1e^mnx terms.
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?
y=c1e^mx + c2xe^mx
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?
y=e^alphax * (c1 cos beta x + c2 sin beta x) where the roots are alpha + i(beta) and alpha - i(beta).
What is the general solution of a DE in the form y’‘+k^2y = 0?
y=c1 cos kx + c2 sin kx
What is the general solution of a DE in the form y’‘-k^2y=0
y=c1e^kx + c2e^-kx
What does it mean if a function is analytic at a point a?
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.
What is an ordinary point of a DE?
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
What is a singular point of a DE?
A point that is not an ordinary point of the differential equation a2(x)y’‘+a1(x)y’+a0(x)y=0?
What is a regular singular point of DE?
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).
What is the method of Frobenius?
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
How do you solve a differential equation using the method of undetermined coefficients and annihilator approach?
- Determine complementary solution yc by solving the LHS as a homogeneous equation.
- Determine the remainder of the terms by determining the annihilator function that will eliminate the RHS (if RHS is nonhomogeneous)
- 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.
- Determine derivative and second derivative of this function as necessary.
- Plug the derivatives of yp into the original equation for y.
- Solve for coefficients, and plug the determined coefficients into the general solution for the particular portion.
What annihilates 1, x, x^2, x^(n-1)?
D^n
What annihilates e^(alpha)x, xe^(alpha)x, x^2e^(alpha)x, x^(n-1)e^(alpha)x?
(D - (alpha)^n
What annihilates e^(alpha)x sin/cos (beta)x, xe^(alpha)x sin/cos (beta)x, etc?
[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
How do you solve a differential equation using variation of parameters?
- Get the equation into standard form y’’ + P(x)y’ + Q(x)y=f(x)
- Determine the complementary solution using the homogeneous part from the LHS
- Y1 = The first non-coefficient function from the complementary solution, Y2 = the second non-coefficient function from the complementary solution.
- Find the Wronskian with Y1 in the first column and Y2 in the second
- Yp = -y1 * int[(y2*f(x))/W] + y2 * int[(y1*f(x)/W]
- Combine complementary solution and particular solution
