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Chapter 4: Q21E (page 164)

First-Order Constant-Coefficient Equations.

  1. Substituting y = ert, find the auxiliary equation for the first-order linear equation ay'+by = 0,whereaandbare constants witha鈮�0.
  2. Use the result of part (a) to find the general solution.

Short Answer

Expert verified

The general solution is y(t)=ce-bat, where c is any constant.

Step by step solution

01

Find the auxiliary equation.

The given differential equation is ay' +by = 0.

If y = ert then y' = rert.

Now substitute the all values in the equation ay'+by = 0, then;

a(rert)+b(ert)=0(ar+b)ert=0ar+b=0

Therefore, the auxiliary equation is (ar+b) = 0

02

Find the general solution.

Find the roots of the auxiliary equation.

(ar+b)=0r=-ba

Thereforerole="math" localid="1655722139481" y(t)=e-bat.

Thus, the general solution is y(t)=ce-bat, where c is any constant.

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