First Order Differential Equations

Linear/Nonlinear

100

The formula for linear first-order derivatives is:

dy/dx+P(x)y=Q(x)

200

The three main differences between linear and nonlinear are:

• y or dx/dy is raised to a power other than 1

• y and dx/dy are multiplied together
• Functions of y such as siny, e^y, or lny appear

300

Solve for y: dx/dy+4y=8 

y=2+Ce^−4x

400

Solve for y: xy'+3y=4x^2-3x

y=4x^2/5-3x/4+C/x^3

Separable/Nonseparable

100

The formula for separable first-order derivatives is:

(1/g(y))dy=f(x)dx

200

What is the difference between separable and non- separable equations:

Separable can have variables separated to one side while non-separable cannot be rearranged into that format. 

300

solve this equation: (dy)/(dx)=x⋅y 

y=Ce^{(x^2)/2}

400

solve this equation: (dy)/(dx)=x/y 

y²=x²+C or y=±√(x²+C) 

Homogeneous

100

The formula for homogenous first-order derivatives is:

dy/dx=f(y/x)

200

An equation is homogenous when:

All the terms are raised to the same degree

300

Solve for y: dy/dx=x+y/x

y=x(ln|x|+C)

400

Solve for y: dy/dx=2x+y/x

y=x(2ln|x|+C)

Exact

100

The formula for exact first-order derivatives is:

 M(x,y)dx + N(x,y)dy=0

200

An equation is exact when:

where the partial derivative of one term with respect to y equals the partial derivative of another term with respect to x

300

solve this equation:(2x+y)dx+(x+3y²)dy=0 

x²+xy+y³=C

400

solve this equation:(3x²+2y)dx+(2x+4y)dy=0 

x³+2xy+2y²=C

Euler's Method

100

The definition of the Euler Method.

What is estimating a differential equation by calculating the tangent line at small intervals.

200

The formula for the Euler method.

What is y_n=y_n-1+hF(x_n-1,y_n-1)?

300

The solution of the differential equation dy/dx=x+y using the Euler Method with the initial condition of y(0)=1 and a step size of 0.1 for the first 3 increments.

What is
y(0.1)= 1.1

y(0.2)= 1.22

y(0.3)= 1.362?

400

The term when Zach hall is slacking off

What is Zachin' it?