Rearrange so that N(y)dy = M(x)dx - Integrate both sides and solve Solve the equation of the form y' = f(x) How can one solve for a second solution of the equation a0(t)y''+a1(t)y'+a2(t)y=0 given its Wronskian W and one of its solutions y₁ How to solve an exact equation M(x)dx + N(y)dy = 0 Solve the equation of the form dy/dx = M(x)/N(y)

Solve the equation of the form dy/dx = M(x)/N(y)

Rearrange so that (1/y)dy = -p(t)dt - Integrate on both sides and solve Solve the equation of the form y' + p(t)y = 0 When solving the equation of the form a0(t)y''+a1(t)y'+a2(t)y=g(t) by Method of Undetermined Coefficients, what particular solution should you guess with if g(t) is a trigonometric function What is the Wronskian of 2 solutions y₁ and y₂ What is cosh(t)

Solve the equation of the form y' + p(t)y = 0

Find u(t) such that u'(t) = u(t)p(t) - Multiply both sides of the equation by u(t) - Simplify and integrate What is the solution of ax²y''+bxy'+cy=0 given that its roots k₁=k₂ What relationship between the equation a0(t)y''+a1(t)y'+a2(t)y=0 and its Wronskian is useful for confirming the validity of its answer What is the solution of ay''+by'+cy=0 given that its roots k₁,₂=a±Bi Solve the equation of the form y' + p(t)y = g(t) using method of integrating factors

Solve the equation of the form y' + p(t)y = g(t) using method of integrating factors

Find the solution of y' + p(t)y = 0 - Replace the unknown constant C with the unknown equation u - Sub homogenous solution into inhomogenous equation and solve for u - Plug u value into homogenous solution to find the inhomogenous solution How do you know if an integrating factor for M(x)dx + N(y)dy = 0 is a function u(y)? What is u(y)? Solve the equation of the form y' + p(t)y = g(t) using method of variation of parameters Solve the equation of the form y' = f(y/x) How do you know if an integrating factor is a function u(xy)? What is u(xy)?

Solve the equation of the form y' + p(t)y = g(t) using method of variation of parameters

Divide both sides by yⁿ - Sub into the equation z = y¹⁻ⁿ and z' = y'*(y¹⁻ⁿ)' - Solve as you would an inhomogenous equation - Sub y¹⁻ⁿ back into general solution Solve the equation of the form y' + p(t)y = g(t)yⁿ Solve the equation of the form dy/dx = M(x)/N(y) What is the solution of ay''+by'+cy=0 given that its roots k₁,₂=a±Bi How to solve an exact equation M(x)dx + N(y)dy = 0

Solve the equation of the form y' + p(t)y = g(t)yⁿ

Sub ux into y to get (ux)' = f(u) - Solve new equation as a seperable equation for x - To find the solution for y, multiply x's solution by u What is the solution of ax²y''+bxy'+cy=0 given that its roots k₁≠k₂ When solving the equation of the form a0(t)y''+a1(t)y'+a2(t)y=g(t) by Method of Undetermined Coefficients, what particular solution should you guess with if g(t) is a trigonometric function Solve the equation of the form y' + p(t)y = g(t) using method of variation of parameters Solve the equation of the form y' = f(y/x)

Solve the equation of the form y' = f(y/x)

Rearrange so that dy = f(x)dx - Integrate both sides and solve What form must you convert a 2nd Order Euler's Equation ax²y''+bxy'+cy=0 into in order to solve Solve the equation of the form y' + p(t)y = 0 When solving the equation of the form a0(t)y''+a1(t)y'+a2(t)y=g(t) by Method of Undetermined Coefficients, what particular solution should you guess with if g(t) is a polynomial Solve the equation of the form y' = f(x)

Solve the equation of the form y' = f(x)

An equation M(x,y)dx + N(x,y)dy = 0 is exact if and only if Mdy = Ndx How to tell if the equation M(x,y)dx + N(x,y)dy = 0 is Exact How to solve an exact equation M(x)dx + N(y)dy = 0 What form is the general solution of a0(t)y''+a1(t)y'+a2(t)y=0 How do you know if an integrating factor is a function u(x)? What is u(x)?

How to tell if the equation M(x,y)dx + N(x,y)dy = 0 is Exact

Integrate M(x,y) by x and N(x,y) by y, but replace the unknown constance C1 and C2 with the unknown equations f and g - Find the values f and g such that both integrals equal eachother - The similar equation + C is the solution What form must you convert a 2nd Order Euler's Equation ax²y''+bxy'+cy=0 into in order to solve How can one solve for a second solution of the equation a0(t)y''+a1(t)y'+a2(t)y=0 given its Wronskian W and one of its solutions y₁ How to solve an exact equation M(x)dx + N(y)dy = 0 Solve the equation of the form dy/dx = M(x)/N(y)

How to solve an exact equation M(x)dx + N(y)dy = 0

If [Ndx - Mdy]/M is a function g(y), then the integrating factor is the function u(y) such that ln(u(y)) = ∫g(y)dy Solve the equation of the form y' = f(x) When solving the equation of the form a0(t)y''+a1(t)y'+a2(t)y=g(t) by Method of Undetermined Coefficients, what particular solution should you guess with if g(t) is of the form Ceᵈᵗ for some constants C,d, and eᵈᵗ can be found in the homogenous equation's solution Solve the equation of the form a0(t)y''+a1(t)y'+a2(t)y=g(t) using the method of variation of parameters How do you know if an integrating factor for M(x)dx + N(y)dy = 0 is a function u(y)? What is u(y)?

How do you know if an integrating factor for M(x)dx + N(y)dy = 0 is a function u(y)? What is u(y)?

MAT244 Review

Flashcard

Learn Mode

Match

Library

Create

Flashcards

Library

Match (Coming Soon)

Calculus - Geometry

user_jevbwl Created by 6 mon ago

Cards in this deck(31)

Rearrange so that N(y)dy = M(x)dx - Integrate both sides and solve

Rearrange so that (1/y)dy = -p(t)dt - Integrate on both sides and solve

Find u(t) such that u'(t) = u(t)p(t) - Multiply both sides of the equation by u(t) - Simplify and integrate

Find the solution of y' + p(t)y = 0 - Replace the unknown constant C with the unknown equation u - Sub homogenous solution into inhomogenous equation and solve for u - Plug u value into homogenous solution to find the inhomogenous solution

Divide both sides by yⁿ - Sub into the equation z = y¹⁻ⁿ and z' = y'*(y¹⁻ⁿ)' - Solve as you would an inhomogenous equation - Sub y¹⁻ⁿ back into general solution

Sub ux into y to get (ux)' = f(u) - Solve new equation as a seperable equation for x - To find the solution for y, multiply x's solution by u

Rearrange so that dy = f(x)dx - Integrate both sides and solve

An equation M(x,y)dx + N(x,y)dy = 0 is exact if and only if Mdy = Ndx

Integrate M(x,y) by x and N(x,y) by y, but replace the unknown constance C1 and C2 with the unknown equations f and g - Find the values f and g such that both integrals equal eachother - The similar equation + C is the solution

If [Ndx - Mdy]/M is a function g(y), then the integrating factor is the function u(y) such that ln(u(y)) = ∫g(y)dy

If -[Ndx - Mdy]/N is a function g(x), then the integrating factor is the function u(x) such that ln(u(x)) = ∫g(x)dy

If [Ndx - Mdy]/[M*x - N*y] is a function g(xy), then the integrating factor is the function u(xy) such that ln(u(t)) - ∫g(t)dt where t=xy

y = (C₁+C₂t)eᵏᵗ

y = C₁eᵏ¹ᵗ+ C₂eᵏ²ᵗ

y=eᵃᵗ(C₁cos(Bt)+ C₂sin(Bt))

y = C₁y₁ + C₂y₂ for some constants C₁,C₂ and solutions y₁,y₂

W[y₁,y₂] = det| y₁ y₂ | | y'₁ y'₂ |

(d/dt)W[y₁,y₂] = (-a1/a0)W[y₁,y₂]

You must convert it into a Indicial Equation: ak² + (b-a)k + c = 0

y = (C₁+C₂ln(x))xᵏ

y = C₁xᵏ¹+ C₂xᵏ²

y=xᵃ(C₁cos(Bln(x))+ C₂sin(Bln(x)))

y = y???? + yₚ, with y???? being the general solution for a0(t)y''+a1(t)y'+a2(t)y=g(t) and yₚ being the particular solution of the original inhomogenous equation

Ax²+Bx+C

Aeᵈᵗ

Ateᵈᵗ

Begin with Ag(t) and change according to findings

Solve for the general solution of a0(t)y''+a1(t)y'+a2(t)y=0 and write it as y = u₁y₁+u₂y₂ for some unknown function u₁ and u₂, and some solutions y₁ and y₂ - Solve for the Wronskian W[y₁,y₂] - Solve for u₁ = -∫g(t)y₂(t)/W dt - Solve for u₂ = ∫g(t)y₁(t)/W dt - Sub u functions into general solution of homogenous equation to find the inhomogenous general solution

y₂ = y₁∫-Wy₁⁻²

Ask Our AI Tutor

Get Instant Help with Your Questions

Need help understanding a concept or solving a problem? Type your question below, and our AI tutor will provide a personalized answer in real-time!

How it works

Ask any academic question, and our AI tutor will respond instantly with explanations, solutions, or examples.

Get Started

Browse questions and discover topic-based flashcards
Practice with engaging flashcards designed for each subject
Strengthen memory with concise, effective learning tools

Discover By Topic

MAT244 Review

Related Decks