LINEAR ALGEBRA AND DIFF.
EQUATIONS: TERMS EXAM 1
QUESTIONS WITH CORRET ANSWERS
differential equation - Answer-an equation involving a function and one or more of its
derivatives (2 types)
ordinary differential equation - Answer-differential equation that involves functions with
only one independent variable
partial differential equation - Answer-a differential equation that involves functions with
more than one independent variable and a partial derivative (i.e. ∂y/∂x)
order of a differential equation - Answer-the highest order of the derivative that appears
solution to a differential equation - Answer-a function or relation that satisfies the
differential equation
-some of these might only work for a particular interval
-if this is a function, we call it explicit, otherwise, it's implicit
direction field (slope field) - Answer-a graph plotting line segments for a DE of the form
dy/dx=f(x,y)
existence and uniqueness theorem - Answer-given the IVP: dy/dx=f(x,y) and y(x₀)=y₀
the IVP (initial value problem) has a unique solition on an open interval containing x₀,
provided f(x,y) and δf/δy are both continuous on a rectangle around (x₀,y₀)
separable differential equation - Answer-given: dy/dx=f(x,y)
f(x,y)=g(x)h(y) (or if the DE can be rearranged into that form)
linear first-order differential equation - Answer-a₁(x)(dy/dx)+a₀(x)y=F(x)
general case: linear first-order DE - Answer-dy/dx+P(x)y=Q(x)
linear first-order DE theorem - Answer-If P(x) and Q(x) are continuous on an interval
(a,b) containing x, then for any choice of initial value y, then the IVP [dy/dx+P(x)y=Q(x),
y(x₀)=y₀] has exactly one solution on (a,b)
exact solution - Answer-M(x,y)dx+N(x,y)dy is exact if there is an f(x,y) such that (δf/δx)
(x,y)=M(x,y) and (δf/δy)(x,y)=N(x,y)
EQUATIONS: TERMS EXAM 1
QUESTIONS WITH CORRET ANSWERS
differential equation - Answer-an equation involving a function and one or more of its
derivatives (2 types)
ordinary differential equation - Answer-differential equation that involves functions with
only one independent variable
partial differential equation - Answer-a differential equation that involves functions with
more than one independent variable and a partial derivative (i.e. ∂y/∂x)
order of a differential equation - Answer-the highest order of the derivative that appears
solution to a differential equation - Answer-a function or relation that satisfies the
differential equation
-some of these might only work for a particular interval
-if this is a function, we call it explicit, otherwise, it's implicit
direction field (slope field) - Answer-a graph plotting line segments for a DE of the form
dy/dx=f(x,y)
existence and uniqueness theorem - Answer-given the IVP: dy/dx=f(x,y) and y(x₀)=y₀
the IVP (initial value problem) has a unique solition on an open interval containing x₀,
provided f(x,y) and δf/δy are both continuous on a rectangle around (x₀,y₀)
separable differential equation - Answer-given: dy/dx=f(x,y)
f(x,y)=g(x)h(y) (or if the DE can be rearranged into that form)
linear first-order differential equation - Answer-a₁(x)(dy/dx)+a₀(x)y=F(x)
general case: linear first-order DE - Answer-dy/dx+P(x)y=Q(x)
linear first-order DE theorem - Answer-If P(x) and Q(x) are continuous on an interval
(a,b) containing x, then for any choice of initial value y, then the IVP [dy/dx+P(x)y=Q(x),
y(x₀)=y₀] has exactly one solution on (a,b)
exact solution - Answer-M(x,y)dx+N(x,y)dy is exact if there is an f(x,y) such that (δf/δx)
(x,y)=M(x,y) and (δf/δy)(x,y)=N(x,y)
