Chapter 4: Q44E (page 200)

Find a second linearly independent solution using reduction of order.
$ty'' + (1 - 2 t) y' + (t - 1) y = 0, t > 0; f (t) = e^{t}$

Short Answer

Expert verified

The second linearly independent solution of the given equation:

$ty'' + (1 - 2 t) y' + (t - 1) y = 0, t > 0; f (t) = e^{t}$ is $y = c_{1} e^{t} + c_{2} e^{t} lnt$ .

Step by step solution

Finding y

Given differential equation is $ty'' + (1 - 2 t) y' + (t - 1) y = 0$ then the standard form of the solution is $y'' + \frac{1 - 2 t}{t} y' + \frac{t - 1}{t} x = 0$ and $f (t) = e^{t}$ is one of the solutions and $p (t) = \frac{1 - 2 t}{t}$ .

Then:

$y_{2} = y_{1} (t) \int \frac{e^{- \int p (t)} dt}{y_{1}^{2} (t)} dt$ .

Simplification

Simplify the above

$\begin{array}{rcl} d y & = & e^{t} \int \frac{e^{- \int \frac{1 - 2 t}{t}} dt}{{(e^{t})}^{2}} dt \\ = & e^{t} \int \frac{e^{2 t - \ln (t)}}{e^{2 t}} dt \\ = & e^{t} \times \int \frac{t^{- 1} \times e^{2 t}}{e^{2 t}} dt \\ y & = & {lnte}^{t} \end{array}$

So, the solution is $y = c_{1} e^{t} + c_{2} e^{t} lnt$ .

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Find a second linearly independent solution using reduction of order.
$ty'' + (1 - 2 t) y' + (t - 1) y = 0, t > 0; f (t) = e^{t}$

Short Answer

Step by step solution

Finding y

Simplification

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Find a second linearly independent solution using reduction of order.ty''+(1-2t)y'+(t-1)y=0,t>0;f(t)=et

Short Answer

Step by step solution

Finding y

Simplification

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Mechanics Maths

Applied Mathematics

Discrete Mathematics

Calculus

Pure Maths

Study anywhere. Anytime. Across all devices.

Find a second linearly independent solution using reduction of order.
$ty'' + (1 - 2 t) y' + (t - 1) y = 0, t > 0; f (t) = e^{t}$