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Separable Differential Equations for AP Calculus

By — McGraw-Hill Professional
Updated on Oct 24, 2011

Practice problems for these concepts can be found at: Applications of Definite Integrals Practice Problems for AP Calculus

General Procedure

  1. Separate the variables: g(y)dy = f(x)dx.
  2. Integrate both sides: g(y)dy = f(x)dx.
  3. Solve for y to get a general solution.
  4. Substitute given conditions to get a particular solution.
  5. Verify your result by differentiating.

Example 1

Given = 4x3 y2 and y(1) = , solve the differential equation.

Step 1.   Separate the variables: dy = 4x3dx.

Step 2.   Integrate both sides:

Step 3.   General solution:

Step 4.   Particular solution:

Step 5.   Verify result by differentiating.

Example 2

Find a solution of the differentiation equation = xsin(x2); y(0) = – 1.

Step 1.   Separate variables: dy = x sin(x2)dx.

Step 2.   Integrate both sides:

Step 3.   Substitute given condition:

Example 3

If = 2x +1 and at x = 0, y ' = – 1, and y = 3, find a solution of the differential equation.

Step 1.   Rewrite

Step 2.   Separate variables: dy ' = (2x +1)dx.

Step 3.   Integrate both sides: dy ' = (2x +1)dx; y ' = x2 + x + C1.

Step 4.   Substitute given condition: At x =0, y ' = – 1; –1 = 0 + 0 + C1 C1 = – 1.

    Thus, y ' =x2 + x – 1.

Step 5.   Rewrite: dy ' = = x2 + x – 1.

Step 6.   Separate variables: dy = (x2 + x – 1)dx.

Step 7.   Integrate both sides: dy = (x2 + x – 1) dx

Step 8.   Substitute given condition: At x = 0, y '= 3; 3 = 0 + 0 – 0 + C2 C2 = 3.

    Therefore,

Step 9.   Verify result by differentiating:

Example 4

Find the general solution of the differential equation

Step 1.   Separate variables:

Step 2.   Integrate both sides: (let u = x2 +1; du = 2x dx)

    In|y|=ln(x2 + 1) + C1.

Step 3.  General Solution: solve for y.

    eln|y| = eln(x2 + 1) + C1
    |y| = eln(x2 + 1) · e C1; |y| = eC1 (x2 + 1)
    y = ± eC1(x 2 +1)
    The general solution is y = C(x2 + 1).

Step 4.   Verify result by differentiating:

    y = C(x2 + 1)

Example 5

Write an equation for the curve that passes through the point (3, 4) and has a slope at any point (x, y) as

Step 1.   Separate variables: 2y dy = (x2 + 1)dx.

Step 2.   Integrate both sides:

Step 3.   Substitute given condition:

    Thus,

Step 4.   Verify the result by differentiating:

Practice problems for these concepts can be found at: Applications of Definite Integrals Practice Problems for AP Calculus

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