APMA 2130 UVA 2.1-2.6
first order linear equation (form) - correct answer ✔✔P(t)y' + Q(t)y = G(t)
Standard form of a first order linear equation - correct answer ✔✔y' + p(t)y = g(t)
Solving a first order linear equation - correct answer ✔✔Use the integrating factor.
1. Put the equation in standard form
2. integrating factor = μ = e^∫p(t)dt
3. Multiply both sides of the equation by the integrating factor
4. d/dt(μy) = μg(t)
5. Integrate both sides
6. Solve for y
Separable equation (form) - correct answer ✔✔M(y)y'(x) = N(x)
Solving a separable equation - correct answer ✔✔1. Put y' in the form of dy/dx
2. Multiply both sides by dx, so you get dx on one side and dy on the other
3. Move stuff around to get it into the form f(x)dx = g(y)dy
4. Integrate both sides
5. Solve
, Homogeneous equation (form) - correct answer ✔✔y' = f(x,y), where f(x,y) can be written as g(v), where
v = y/x
Solving a homogeneous equation - correct answer ✔✔1. Since v = y/x, y = vx
2. Find y' in terms of v and x
3. Substitute in y' and y, in terms of v and x
4. You should get a separable equation, in terms of v and x. Solve for v.
5. plug in y/x for v
6. Solve for y
Exponential growth - correct answer ✔✔Where rate of population growth is directly proportional to
population size
Exponential growth (form of differential equation) - correct answer ✔✔y' = ky, where k>0
Exponential decay (form of differential equation) - correct answer ✔✔y' = ky, where k < 0
Coefficient functions - correct answer ✔✔Standard form is y'(t) + p(t)y = g(t)
p(t) and g(t) are the coefficient functions
Uniqueness theorem for a linear initial value problem - correct answer ✔✔Given y'(t) + p(t)y = g(t), y(t0)
= y0, then:
first order linear equation (form) - correct answer ✔✔P(t)y' + Q(t)y = G(t)
Standard form of a first order linear equation - correct answer ✔✔y' + p(t)y = g(t)
Solving a first order linear equation - correct answer ✔✔Use the integrating factor.
1. Put the equation in standard form
2. integrating factor = μ = e^∫p(t)dt
3. Multiply both sides of the equation by the integrating factor
4. d/dt(μy) = μg(t)
5. Integrate both sides
6. Solve for y
Separable equation (form) - correct answer ✔✔M(y)y'(x) = N(x)
Solving a separable equation - correct answer ✔✔1. Put y' in the form of dy/dx
2. Multiply both sides by dx, so you get dx on one side and dy on the other
3. Move stuff around to get it into the form f(x)dx = g(y)dy
4. Integrate both sides
5. Solve
, Homogeneous equation (form) - correct answer ✔✔y' = f(x,y), where f(x,y) can be written as g(v), where
v = y/x
Solving a homogeneous equation - correct answer ✔✔1. Since v = y/x, y = vx
2. Find y' in terms of v and x
3. Substitute in y' and y, in terms of v and x
4. You should get a separable equation, in terms of v and x. Solve for v.
5. plug in y/x for v
6. Solve for y
Exponential growth - correct answer ✔✔Where rate of population growth is directly proportional to
population size
Exponential growth (form of differential equation) - correct answer ✔✔y' = ky, where k>0
Exponential decay (form of differential equation) - correct answer ✔✔y' = ky, where k < 0
Coefficient functions - correct answer ✔✔Standard form is y'(t) + p(t)y = g(t)
p(t) and g(t) are the coefficient functions
Uniqueness theorem for a linear initial value problem - correct answer ✔✔Given y'(t) + p(t)y = g(t), y(t0)
= y0, then:
