Differential Equations Intro Cheat Sheet

A printable reference covering differential equations, slope fields, separable equations, initial values, and exponential growth and decay for grades 11-12.

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Differential equations connect a function to its rate of change, making them useful for modeling motion, population growth, cooling, and many other real situations. This cheat sheet introduces the notation, solution ideas, and common first-order forms students meet at the start of calculus-based differential equations. It helps students recognize what a differential equation is asking and choose a basic solving strategy. The main ideas are interpreting $\frac{dy}{dx}$ , using slope fields, solving separable equations, and applying initial conditions. A general solution contains a constant such as $C$ , while a particular solution uses information like $y(0)=5$ to find that constant. Many beginner models use exponential growth or decay, written as $\frac{dy}{dt}=ky$ with solution $y=Ce^{kt}$ .

Key Facts

A differential equation is an equation involving an unknown function and one or more derivatives, such as $\frac{dy}{dx}=3x^2$ .
The order of a differential equation is the highest derivative present, so $\frac{d^2y}{dx^2}+y=0$ is second order.
A general solution includes an arbitrary constant, such as $y=x^3+C$ for $\frac{dy}{dx}=3x^2$ .
A particular solution uses an initial condition, such as $y(0)=2$ , to find the value of the constant $C$ .
A separable differential equation can be written as $g(y)\,dy=f(x)\,dx$ before integrating both sides.
To solve $\frac{dy}{dx}=f(x)g(y)$ , rewrite it as $\frac{1}{g(y)}\,dy=f(x)\,dx$ and integrate.
The exponential model $\frac{dy}{dt}=ky$ has solution $y=Ce^{kt}$ , where $k>0$ gives growth and $k<0$ gives decay.
A slope field shows short line segments with slope $\frac{dy}{dx}$ at points $(x,y)$ , giving a visual picture of solution curves.

Vocabulary

Differential equation: An equation that relates an unknown function to one or more of its derivatives.
General solution: A family of solutions containing an arbitrary constant, often written with $C$ .
Particular solution: One specific solution found by using an initial condition to determine the constant.
Initial condition: A given value of the function, such as $y(0)=4$ , used to select one solution from a family.
Separable equation: A differential equation that can be rearranged so all $y$ terms are with $dy$ and all $x$ terms are with $dx$ .
Slope field: A diagram of small line segments showing the slope of solutions at many points in the plane.

Common Mistakes to Avoid

Forgetting the constant $C$ after integrating is wrong because an indefinite integral represents a family of functions, not just one function.
Using the initial condition too early is wrong because you usually need the general solution first before substituting values like $x=0$ and $y=2$ .
Separating variables incorrectly is wrong because equations such as $\frac{dy}{dx}=x+y$ cannot be rewritten as $g(y)\,dy=f(x)\,dx$ by simple algebra.
Mixing up growth and decay is wrong because $k>0$ in $y=Ce^{kt}$ increases over time, while $k<0$ decreases over time.
Drawing slope fields with random segment directions is wrong because each segment must match the slope given by $\frac{dy}{dx}$ at that exact point.

Practice Questions

1 Find the general solution of $\frac{dy}{dx}=6x-4$ .
2 Solve $\frac{dy}{dt}=3y$ with the initial condition $y(0)=5$ .
3 Solve the separable differential equation $\frac{dy}{dx}=xy$ with $y(0)=2$ .
4 Explain how a slope field can show whether solutions to $\frac{dy}{dt}=ky$ are growing or decaying without solving the equation.