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CalculusBy Educator Today

C30 · Unit C7 Differential equations

Separation and initial conditions

CoreReview editionNot yet human-reviewed
Time
About 45 minutes, untimed
Course framework reference
AP 7.6–7.7
Builds on
Part of

Progress on this lesson counts in every pathway listed.

Understand the idea

For y minutes equals g(x)h(y), separate as dy over h(y) equals g(x)dx where h(y) is not equal to 0 and integrate both sides. Solve for y where possible, then impose the initial condition. Dividing by h(y) can omit constant equilibrium solutions at its zeros, so check them separately. A formula’s validity interval must respect both the equation and any singularities.

Worked example

For y minutes equals xy and y(0) equals 2, separation gives ln|y| equals x squared /2+C, so y equals Ce^(x squared /2). The initial value gives C equals 2. The equilibrium y equals 0 also solves the original equation but not this initial-value problem.

Explore the mathematics

Separation and initial conditions

Separate a differential equation while checking any solutions excluded by division.

Setting 3 of 5

Scroll sideways to see the full diagram.

Can you separate the variables in this rate law?xy1234dy/dx=y, y(0)=1

Predict before continuing

Can you separate the variables in this rate law?

Read the complete explanation
  1. Can you separate the variables in this rate law?
  2. For y≠0, write dy/y=dx; integrate both sides.
  3. ln|y|=x+C, hence y=Aeˣ. Apply the initial condition to obtain A=1.
  4. The zero solution y=0 must be checked separately because division by y excludes it; it does not satisfy this nonzero initial condition.
See the idea: Separation and initial conditions
See the idea: Separation and initial conditionsThe positive solution 2e^(x²/2) passes through (0,2) and satisfies y′=xy. The horizontal equilibrium y=0 also solves the differential equation, but not this initial condition.xy-10-0.51020.5314initial value 2

The positive solution 2e^(x²/2) passes through (0,2) and satisfies y′=xy. The horizontal equilibrium y=0 also solves the differential equation, but not this initial condition.

Discuss: explain what each label represents, then draw a related example with different values.

A common misconception

Indefinite integration requires a constant. Do not lose equilibria by dividing by a potentially zero dependent variable.

Investigate and explain

Compare a slope field with the explicit solution family for y minutes equals xy. Discuss where the curves rise or fall.
Identify one step in the worked example, name the mathematical fact that permits it, and state its domain conditions.