Chapter 4: Q33E (page 200)

Use Abel's formula (Problem) to determine (up to a constant multiple) the Wronskian of two solutions on $(0, \infty)$ to $ty'' + (t - 1) y' + 3 y = 0$ .

Short Answer

Expert verified

The solution for the given equation $ty'' + (t - 1) y' + 3 y = 0$ is $W [y_{1}, y_{2}] = {cte}^{- t}$ .

Step by step solution

Divide byon both sides

Given differential equation is:

$\begin{array}{rcl} ty'' + (t - 1) y' + 3 y & = & 0 \\ y'' + \frac{t - 1}{t} y' + \frac{3}{t - 1} y & = & 0 \end{array}$

Here role="math" localid="1664188072730" $p (t) = \frac{t - 1}{t} = 1 - \frac{1}{t}$

Use Abel’s formula

Then from problem number 32

$\begin{array}{rcl} W [y_{1}, y_{2}] & = & {ce}^{\int_{t_{0}}^{t} - (1 - \frac{1}{t}) dt} \\ = & {ce}^{\int_{t_{0}}^{t} (\frac{1}{t} - 1) dt} \\ = & {ce}^{(lnt - t)} \\ = & {ce}^{lnt} e^{- t} \\ = & {cte}^{- t} \end{array}$

$W [y_{1}, y_{2}] = {cte}^{- t}$

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Use Abel's formula (Problem) to determine (up to a constant multiple) the Wronskian of two solutions on $(0, \infty)$ to $ty'' + (t - 1) y' + 3 y = 0$ .

Short Answer

Step by step solution

Divide byon both sides

Use Abel’s formula

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Use Abel's formula (Problem) to determine (up to a constant multiple) the Wronskian of two solutions on(0,∞)toty''+(t-1)y'+3y=0.

Short Answer

Step by step solution

Divide byon both sides

Use Abel’s formula

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Applied Mathematics

Theoretical and Mathematical Physics

Statistics

Geometry

Mechanics Maths

Study anywhere. Anytime. Across all devices.

Use Abel's formula (Problem) to determine (up to a constant multiple) the Wronskian of two solutions on $(0, \infty)$ to $ty'' + (t - 1) y' + 3 y = 0$ .