4.6 - VARIATION OF PARAMETERS

MATH 2584 - ELEMENTARY DIFFERENTIAL EQUATIONS - FALL 2017

To solve y′′ + P (x)y′ + Q(x)y = f(x), we first find a fundamental set of solutions y1 and y2for the associated homogeneous equation.

Then we guess that our particular solution takes the form yp = u1(x)y1(x) + u2(x)y2(x).

Calculations:

d

dx[y1u

′1 + y2u

′2] + P [y1u

′1 + y2u

′2] + y′1u

′1 + y′2u

′2 = f(x).

This is a difficult equation to solve, so we make the simplifying assumption

y1u′1 + y2u

′2 = 0.

This leaves us withy′1u

′1 + y′2u

′2 = f(x).

Now we have two equations and two unknowns, so we can solve for u′1 and u′

2.

Calculations:

We will have

u′1 =

−y2f(x)

y1y′2 − y2y′1and u′

2 =y1f(x)

y1y′2 − y2y′1.

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To solve a second order linear equation by variation of parameters:

(1) Put the equation in standard form y′′ + P (x)y′ + Q(x)y = f(x).(2) Find the complementary solution yc = c1y1 + c2y2.(3) Compute the Wronskian

W =

∣∣∣∣ y1 y2y′1 y′2

∣∣∣∣(4) Compute

W1 =

∣∣∣∣ 0 y2f(x) y′2

∣∣∣∣ and W2 =

∣∣∣∣ y1 0y′1 f(x)

∣∣∣∣ .(5) Integrate u′

1 = W1

Wand u′

2 = W2

W.

(6) Set yp = u1y1 + u2y2.

Examples:

(1) Solve y′′ + y = tanx by variation of parameters.

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(2) Solve y′′ − 2y′ + y = ex

1+x2 by variation of parameters.

Homework: (eighth edition) page 161-162, #1, 5, 15, 17(ninth edition) page 165, #1, 5, 15, 17

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