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Calculus Differential equations (introductory)

Separable differential equations

20 practice questions 0 video lessons Theory + worked examples

Separable Differential Equations

California Calculus • Standard 27.0 • Differential Equations

Separable Differential Equations is the opening topic of Differential Equations in the California Calculus Standards. It is aligned to Standard 27.0, which requires students to know the techniques of solution of selected elementary differential equations.

A separable differential equation can be written \(\dfrac{dy}{dx}=g(x)h(y)\), so the variables move to opposite sides and each side is integrated; an initial condition fixes the constant.

California Calculus › Differential Equations › Separable Differential Equations  —  Standard 27.0

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Theory

A separable differential equation can be written so all the \(y\)'s are on one side and all the \(x\)'s on the other. Then you integrate both sides. The result is a family of curves; an initial condition selects one.

A differential equation relates a function to its derivative. It is separable when it can be written

\[\dfrac{dy}{dx}=g(x)\,h(y),\]

so that all the \(y\)-parts move to one side and all the \(x\)-parts to the other. You then integrate both sides.

The general solution contains a constant \(C\) — a whole family of curves. An initial condition \(y(x_0)=y_0\) pins down \(C\), giving the particular solution.

Key idea: the special equation \(\dfrac{dy}{dx}=ky\) is separable and gives \(y=A e^{kx}\) — the exponential model behind growth and decay.
A family of solution curves differing by the constant The general solution of a differential equation is a whole family of curves; the constant of integration shifts one curve to another. x y C=1 C=0 C=-1
The general solution is a family of curves (one per \(C\)).
A single particular solution through a given point An initial condition picks out one curve from the family, the particular solution passing through the given point. x y initial point
An initial condition selects one particular solution.

Separate, then integrate:

\[\dfrac{dy}{h(y)}=g(x)\,dx\ \Rightarrow\ \int\dfrac{dy}{h(y)}=\int g(x)\,dx\]
separate the variables and integrate both sides
\[\dfrac{dy}{dx}=ky\ \Rightarrow\ y=A e^{kx}\]
dy dx equals ky gives y equals A e to the kx
Add \(C\) once. A single constant of integration on one side is enough; use the initial condition to find it.

How to solve a separable equation

  1. Separate the variables: all \(y\) (and \(dy\)) on the left, all \(x\) (and \(dx\)) on the right.
  2. Integrate both sides, adding \(+C\).
  3. Solve for \(y\) if possible, and use an initial condition to find \(C\).
Example 1 — Direct integration
Solve \(\dfrac{dy}{dx}=2x\) with \(y(0)=1\).
Solution

Integrate both sides, then use the initial condition.

\(y\)\(=\)\(\int 2x\,dx=x^2+C\)
\(y(0)=1\)\(\Rightarrow\)\(C=1\)

So \(y=x^2+1\).

solution y equals x squared plus 1
Example 2 — Separate the variables
Solve \(\dfrac{dy}{dx}=2xy\).
Solution

Divide by \(y\), integrate both sides.

\(\dfrac{dy}{y}\)\(=\)\(2x\,dx\)
\(\ln|y|\)\(=\)\(x^2+C\)
\(y\)\(=\)\(A e^{x^2}\)
solution y equals A e to the x squared
Example 3 — An implicit answer
Solve \(\dfrac{dy}{dx}=\dfrac{x}{y}\).
Solution

Multiply by \(y\,dx\), integrate.

\(y\,dy\)\(=\)\(x\,dx\)
\(\dfrac{y^2}{2}\)\(=\)\(\dfrac{x^2}{2}+C\)
\(y^2\)\(=\)\(x^2+C_1\)
solution y squared equals x squared plus a constant
Example 4 — The dy/dx = ky model
Solve \(\dfrac{dy}{dx}=ky\).
Solution

Separate and integrate; the log exponentiates.

\(\dfrac{dy}{y}\)\(=\)\(k\,dx\)
\(\ln|y|\)\(=\)\(kx+C\)
\(y\)\(=\)\(A e^{kx}\)
solution y equals A e to the kx

Common pitfalls

Every \(y\) must leave the right side. You cannot integrate \(\int h(y)\,dx\) — separate first so each side has one variable.
Do not forget \(+C\). The constant is what an initial condition later determines.
Some answers stay implicit. \(y^2=x^2+C\) is a valid solution; you need not always solve for \(y\).

Frequently asked questions

What is a separable differential equation?

One that can be written \(\dfrac{dy}{dx}=g(x)h(y)\), so the variables separate onto opposite sides and each side can be integrated.

How do you solve a separable differential equation?

Separate the variables, integrate both sides (with \(+C\)), then solve for \(y\) and use any initial condition to find the constant.

What is the difference between the general and particular solution?

The general solution contains the constant \(C\) (a family of curves); the particular solution uses an initial condition to fix \(C\) to one curve.

What does dy/dx = ky solve to?

\(y=A e^{kx}\). This exponential is the model for growth (\(k>0\)) and decay (\(k<0\)).

More in Differential equations (introductory)
Exponential growth and decay models
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