Differential Equations

Question 1

What is a DE with only 1 independent variable?

Answer 1

Ordinary Differential Equation (ODE)

Question 2

What is a DE with more than one independent variable?

Answer 2

Partial Differential Equation (PDE)

Question 3

What is meant when a question asks for the “order” of a DE?

Answer 3

The highest derivative in the equation.

Question 4

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 _______.

Answer 4

Linear

Question 5

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

Answer 5

c=-1
y=1/(x^2 - 1)

Question 6

How do you find critical points from an equation?

Answer 6

Take derivative and set it equal to zero, then solve for variable values.

Question 7

Consider (dy/dx)=-y^(2) - y^(3), find the critical points.

Answer 7

y=0
y=1

Question 8

What is a vertical line with critical points marked along it with arrows in between the points to indicate increasing/decreasing?

Answer 8

Phase Portrait

Question 9

Given (dy/dx)=x^(4) find the solution in standard form.

Answer 9

y=(1/5)x^(5) + C

Question 10

Given (x+1)(dy/dx) + y = ln(x), y(1)=10, find the solution in standard form.

Answer 10

y=(xln(x)-x+21)/(x+1)

Question 11

Given M(x,y)dx+N(x,y)dy=0, the DE is exact if and only if _____________.

Answer 11

dM/dy=dN/dx

Question 12

What is an equation in the form
dy/dx + P(x)y = f(x)?

Answer 12

A linear equation

Question 13

What is the first step of solving a linear equation in the form dy/dx + P(x)y = f(x) for y?

Answer 13

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

Question 14

solve x^(2)y’+xy=3

Answer 14

y=(3ln(x)/x)+(C/x)

Question 15

solve x(dy/dx) - y = x^(2)sin(x)

Answer 15

y=(e^(x))/(2x^(2)) +C/(x^2)e^(X)

Question 16

Solve the exact equation
(siny-ysinx)dx+(cosx+xcosy-y)=0

Answer 16

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)

Question 17

How does one make a nonexact DE exact?

Answer 17

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.

Question 18

Determine if the equation (7x+5y)dx+(5x-8y^3)dy=0 is exact. If it is, solve it.

Answer 18

Is exact

F(x,y)=(7/2)x^(2) + 5xy - 2y^(4)

Question 19

Determine if the equation (yln(y) - e^(-xy))dx + ((1/y) + xln(y))dy = 0 is exact. If it is, solve it.

Answer 19

Not exact

Question 20

Find the integration constant u of the non-exact equation
(2y^2 + 6x)dx + (2xy)dy = 0 and solve it.

Answer 20

ux=x

2x^3 + (x^2)(y^2) = C

Question 21

What is the equation dA(t)/dt = ? = R(in) - R(out) used for?

Answer 21

Mixture of 2 solutions

Question 22

y1(x)=x^2 is a solution of
x^(2)y’’ + 2xy’ - 6y = 0
find the second solution y2(x)

Answer 22

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

Question 23

How does one solve a homogeneous linear equation with constant coefficients?

Answer 23

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 -