DIFFERENTIAL EQUATIONS STUDY
QUESTIONS WITH COMPLETE
SOLUTIONS GRADED A+
What is the existence and uniqueness thm for non linear DEs? - Answer-Let f(t,y) and
partial[f(t,y)]/partial y be continuous on some rectangular region a < t < b, c < y < d.
Let (t0, y0) be a point in the rectangular region a < t < b, c < y < d
Then there exists an h > 0 such that the initial value problem
y' + p(t) y = q(t)
y(t0) = y0
has a unique solution y = f(t) defined on the interval t0-h < t < t0+h
When is a DE separable? - Answer-When the variables of the equation can be
separated to different sides of the equation. f(t,y) = g(t)/h(y) or f(t,y) = g(t) h(y) which can
be rewritten as g(t) dt = h(y) dy
What is the difference between ODEs and PDEs? - Answer-If there are any partial
derivatives, then the equation is a partial differential equation. If there is more than one
independent variable, then the equation is a partial differential equation.
What is a Linear DE? - Answer-Ay'+Bxy=0 is linear in the variable y.
First order linear equations have what form? - Answer-P(t)y' + Q(t)y = G(t)
Integrating Factor for First Order Linear Eqs - Answer-Consider the function µ(t) =
e^∫p(t)dt
d(µ(t))/dt = p(t) * µ(t)
so µ'(t) = p(t)µ(t) when µ(t) = e^∫p(t)dt
Integrating Factor is e^∫p(t)dt
Find the independent and dependent variables: f(x, y', y'', y'''). - Answer-x is the
independent variable, y is the dependent variable
How do we study Dif Eqs? - Answer-Analytically: We find an explicit or implicit function y
that satisfies the dif eq.
Numerically: Approximate the values of the solution at certain points.
Visually: Sketch solutions (integral curves) and create a slope(direction) field.
What we mean by y is a solution to the DE y' = f(t,y) on some interval (a,b)? - Answer-y
is a solution to a DE if:
1) y is differentiable on the interval (a,b)
2) y satisfies the equation y'=f(t,y)
QUESTIONS WITH COMPLETE
SOLUTIONS GRADED A+
What is the existence and uniqueness thm for non linear DEs? - Answer-Let f(t,y) and
partial[f(t,y)]/partial y be continuous on some rectangular region a < t < b, c < y < d.
Let (t0, y0) be a point in the rectangular region a < t < b, c < y < d
Then there exists an h > 0 such that the initial value problem
y' + p(t) y = q(t)
y(t0) = y0
has a unique solution y = f(t) defined on the interval t0-h < t < t0+h
When is a DE separable? - Answer-When the variables of the equation can be
separated to different sides of the equation. f(t,y) = g(t)/h(y) or f(t,y) = g(t) h(y) which can
be rewritten as g(t) dt = h(y) dy
What is the difference between ODEs and PDEs? - Answer-If there are any partial
derivatives, then the equation is a partial differential equation. If there is more than one
independent variable, then the equation is a partial differential equation.
What is a Linear DE? - Answer-Ay'+Bxy=0 is linear in the variable y.
First order linear equations have what form? - Answer-P(t)y' + Q(t)y = G(t)
Integrating Factor for First Order Linear Eqs - Answer-Consider the function µ(t) =
e^∫p(t)dt
d(µ(t))/dt = p(t) * µ(t)
so µ'(t) = p(t)µ(t) when µ(t) = e^∫p(t)dt
Integrating Factor is e^∫p(t)dt
Find the independent and dependent variables: f(x, y', y'', y'''). - Answer-x is the
independent variable, y is the dependent variable
How do we study Dif Eqs? - Answer-Analytically: We find an explicit or implicit function y
that satisfies the dif eq.
Numerically: Approximate the values of the solution at certain points.
Visually: Sketch solutions (integral curves) and create a slope(direction) field.
What we mean by y is a solution to the DE y' = f(t,y) on some interval (a,b)? - Answer-y
is a solution to a DE if:
1) y is differentiable on the interval (a,b)
2) y satisfies the equation y'=f(t,y)
