Chapter 4: Q21E (page 164)

First-Order Constant-Coefficient Equations.
Substituting y = e^rt, find the auxiliary equation for the first-order linear equation ay'+by = 0,whereaandbare constants with $a \neq 0$ .
Use the result of part (a) to find the general solution.

Short Answer

Expert verified

The general solution is $y (t) = c e^{- \frac{b}{a} t}$ , where c is any constant.

Step by step solution

Find the auxiliary equation.

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

If y = e^rt then y' = re^rt.

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

$\begin{array}{rcl} a (r e^{r t}) + b (e^{r t}) & = & 0 \\ (a r + b) e^{r t} & = & 0 \\ a r + b & = & 0 \end{array}$

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

Find the general solution.

Find the roots of the auxiliary equation.

$(a r + b) = 0 r = - \frac{b}{a}$

Thereforerole="math" localid="1655722139481" $y (t) = e^{- \frac{b}{a} t}$ .

Thus, the general solution is $y (t) = c e^{- \frac{b}{a} t}$ , where c is any constant.

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First-Order Constant-Coefficient Equations.
Substituting y = e^rt, find the auxiliary equation for the first-order linear equation ay'+by = 0,whereaandbare constants with $a \neq 0$ .
Use the result of part (a) to find the general solution.

Short Answer

Step by step solution

Find the auxiliary equation.

Find the general solution.

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First-Order Constant-Coefficient Equations.Substituting y = ert, find the auxiliary equation for the first-order linear equation ay'+by = 0,whereaandbare constants witha≠0.Use the result of part (a) to find the general solution.

Short Answer

Step by step solution

Find the auxiliary equation.

Find the general solution.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Discrete Mathematics

Geometry

Logic and Functions

Mechanics Maths

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

First-Order Constant-Coefficient Equations.
Substituting y = e^rt, find the auxiliary equation for the first-order linear equation ay'+by = 0,whereaandbare constants with $a \neq 0$ .
Use the result of part (a) to find the general solution.