Differential Equations Flashcards
What is a DE with only 1 independent variable?
Ordinary Differential Equation (ODE)
What is a DE with more than one independent variable?
Partial Differential Equation (PDE)
What is meant when a question asks for the “order” of a DE?
The highest derivative in the equation.
When a DE’s independent variable (and all of its derivatives) are to the power of 1 and are not inside of trig functions, it is considered _______.
Linear
y=1/(x^2 + c) is a family of solutions for the first order DE y’+2xy^2=0. Find the first order IVP given y(2)=1/3
c=-1
y=1/(x^2 - 1)
How do you find critical points from an equation?
Take derivative and set it equal to zero, then solve for variable values.
Consider (dy/dx)=-y^(2) - y^(3), find the critical points.
y=0
y=1
What is a vertical line with critical points marked along it with arrows in between the points to indicate increasing/decreasing?
Phase Portrait
Given (dy/dx)=x^(4) find the solution in standard form.
y=(1/5)x^(5) + C
Given (x+1)(dy/dx) + y = ln(x), y(1)=10, find the solution in standard form.
y=(xln(x)-x+21)/(x+1)
Given M(x,y)dx+N(x,y)dy=0, the DE is exact if and only if _____________.
dM/dy=dN/dx
What is an equation in the form
dy/dx + P(x)y = f(x)?
A linear equation
What is the first step of solving a linear equation in the form dy/dx + P(x)y = f(x) for y?
e^(integral(P(x)dx)
then multiply by the solution of that to both sides where (e^(integral(P(x)dx))[dy/dx + P(x)y] is equal to (e^(integral(P(x)dx))y because of
integral(xy’ + y)=xy
by reverse product rule
solve x^(2)y’+xy=3
y=(3ln(x)/x)+(C/x)
solve x(dy/dx) - y = x^(2)sin(x)
y=(e^(x))/(2x^(2)) +C/(x^2)e^(X)
Solve the exact equation
(siny-ysinx)dx+(cosx+xcosy-y)=0
F(x,y)=integral(M(x,y)dx+g(y))
M=dF/dx=siny-ysinx
F(x,y)=xsiny+ycosx+g(y)
N=dF/dy=d/dy[xsiny+ycosx+g(y)]
N=cosx+xcosy-y
g’(y)=-y
F(x,y)=xsiny+ycosx-(1/2)y^(2)
How does one make a nonexact DE exact?
Multiply DE by an integrating factor u(x,y) to make ~M and ~N and make the statement (d~M)/dy=(d~N)/dx true.
Determine if the equation (7x+5y)dx+(5x-8y^3)dy=0 is exact. If it is, solve it.
Is exact
F(x,y)=(7/2)x^(2) + 5xy - 2y^(4)
Determine if the equation (yln(y) - e^(-xy))dx + ((1/y) + xln(y))dy = 0 is exact. If it is, solve it.
Not exact
Find the integration constant u of the non-exact equation
(2y^2 + 6x)dx + (2xy)dy = 0 and solve it.
ux=x
2x^3 + (x^2)(y^2) = C
What is the equation dA(t)/dt = ? = R(in) - R(out) used for?
Mixture of 2 solutions
y1(x)=x^2 is a solution of
x^(2)y’’ + 2xy’ - 6y = 0
find the second solution y2(x)
standard form: y’’ + (2/x)y’ - (6/x^(2))y = 0
P = 2/x
y2 = u(x)y1(x)
u(x)= int[(e^(-int[P]))/(y1^2)]
y2(x) = u(x)y1(x) = (-x^(-3))/5
How does one solve a homogeneous linear equation with constant coefficients?
Treat y’’ as m^2, y’ as m, and y becomes 1. factor and plug into the forms as follows:
(m(+/-)a)^2 = 0 => c1e^(mx) + c2xe^(mx)
(m(+/-)a)(m(+/-)b) = 0 => c1e^(m1x) + c2e^(m2x)
quadratic formula: imaginary numbers become cos for + and sin for -
