Chapter 4: Q52 E (page 201)

The reduction of order formula $(13)$ can also be derived from Abel’s identity (Problem 32). Let $f (t)$ be a nontrivial solution toanda second linearly independent solution. Show that ${(\frac{y}{f})}^{'} = \frac{W [f, y]}{f^{2}}$ and then use Abel's identity for the Wronskian $W [f, y]$ to obtain the reduction of order formula.

Short Answer

Expert verified

The solution of the given equation is $y = f \int \frac{- e^{\int p (t) dt}}{f^{2}} dt$ .

Step by step solution

Using the formula

From the given ${(\frac{y}{f})}^{'} = \frac{fy' - f' y}{f^{2}}$ and we know $w [f, y] = fy' - f' y$

${(\frac{y}{f})}^{'} = \frac{W [f, y]}{f^{2}}$

Hence proved.

Integration

Now integrate on both sides;

$\int {(\frac{y}{f})}^{'} dt = \int \frac{W [f, y]}{f^{2}} dt \frac{y}{f} = \int \frac{- e^{\int p (t) dt}}{f^{2}} dt y = f \int \frac{- e^{\int p (t) dt}}{f^{2}} dt$

role="math" $\int {(\frac{y}{f})}^{'} dt = \int \frac{W [f, y]}{f^{2}} dt \frac{y}{f} = \int \frac{- e^{\int p (t) dt}}{f^{2}} dt y = f \int \frac{- e^{\int p (t) dt}}{f^{2}} dt$

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The reduction of order formula 13can also be derived from Abel’s identity (Problem 32). Letftbe a nontrivial solution toanda second linearly independent solution. Show thatyf'=W[f,y]f2and then use Abel's identity for the WronskianWf,yto obtain the reduction of order formula.

Short Answer

Step by step solution

Using the formula

Integration

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Discrete Mathematics

Calculus

Geometry

Statistics

Pure Maths

Study anywhere. Anytime. Across all devices.