First-Order ODE Solution Methods Reference Cheat Sheet

A printable reference covering separable equations, linear integrating factors, exact equations, slope fields, and equilibrium solutions for grades 11-12.

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First-order ordinary differential equations describe how a quantity changes when its rate of change depends on the variable, the quantity, or both. This cheat sheet helps students recognize common first-order ODE forms and choose an efficient solution method. It is useful for calculus, AP Calculus enrichment, and introductory differential equations practice. The main goal is to connect the structure of an equation to the method that solves it.

Key Facts

A separable differential equation can be written as $\frac{dy}{dx}=g(x)h(y)$ , then solved by $\frac{1}{h(y)}\,dy=g(x)\,dx$ and integrating both sides.
A first-order linear differential equation has the form $\frac{dy}{dx}+P(x)y=Q(x)$ .
The integrating factor for $\frac{dy}{dx}+P(x)y=Q(x)$ is $\mu(x)=e^{\int P(x)\,dx}$ .
After multiplying a linear equation by $\mu(x)$ , the left side becomes $\frac{d}{dx}[\mu(x)y]=\mu(x)Q(x)$ .
An exact differential equation has the form $M(x,y)\,dx+N(x,y)\,dy=0$ and is exact when $\frac{\partial M}{\partial y}=\frac{\partial N}{\partial x}$ .
For an exact equation, the solution is $F(x,y)=C$ , where $F_x=M$ and $F_y=N$ .
An equilibrium solution occurs when $\frac{dy}{dx}=0$ , so the solution is a constant function $y=c$ .
An initial condition such as $y(x_0)=y_0$ is used after finding the general solution to determine the constant $C$ .

Vocabulary

First-order ODE: A differential equation involving an unknown function and its first derivative, such as $\frac{dy}{dx}=f(x,y)$ .
Separable equation: A differential equation whose variables can be separated into the form $A(y)\,dy=B(x)\,dx$ before integration.
Linear equation: A first-order equation that can be written as $\frac{dy}{dx}+P(x)y=Q(x)$ .
Integrating factor: A function $\mu(x)=e^{\int P(x)\,dx}$ used to turn a linear ODE into a product derivative.
Exact equation: An equation $M(x,y)\,dx+N(x,y)\,dy=0$ for which a potential function $F(x,y)$ satisfies $dF=M\,dx+N\,dy$ .
Slope field: A graph showing small line segments with slope $\frac{dy}{dx}=f(x,y)$ at many points to visualize solution curves.

Common Mistakes to Avoid

Separating variables incorrectly, because terms involving $y$ must stay with $dy$ and terms involving $x$ must stay with $dx$ before integrating.
Forgetting the constant of integration, because solving $\int A(y)\,dy=\int B(x)\,dx$ requires a constant $C$ on one side.
Using the wrong integrating factor, because $\mu(x)=e^{\int P(x)\,dx}$ only works after the equation is in the form $\frac{dy}{dx}+P(x)y=Q(x)$ .
Calling an equation exact without checking, because exactness requires $\frac{\partial M}{\partial y}=\frac{\partial N}{\partial x}$ .
Applying the initial condition too early, because it should usually be used after finding the general solution or implicit solution.

Practice Questions

1 Solve the separable differential equation $\frac{dy}{dx}=3x^2y$ with initial condition $y(0)=2$ .
2 Solve the linear differential equation $\frac{dy}{dx}+2y=e^x$ .
3 Determine whether $(2xy+3)\,dx+(x^2+4y)\,dy=0$ is exact, and if it is exact, find the implicit solution.
4 For the equation $\frac{dy}{dx}=y(4-y)$ , explain what the equilibrium solutions are and describe which one is stable using the sign of $\frac{dy}{dx}$ .

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