Chapter 12: Q13P (page 612)

Usingverify $(22 .19)$ and $(22.20)$ also find $L_{3} (x)$ and $L_{4} (x)$ .

Short Answer

Expert verified

The value of the $L_{3} (x)$ and $L_{4} (x)$ is given by :

$\begin{array}{l} L_{3} (x) = 1 - x + \frac{3 x^{2}}{2} - \frac{x^{3}}{6} \\ L_{4} (x) = 1 - 4 x + 3 x^{2} - \frac{2 x^{3}}{3} + \frac{x^{4}}{24} \end{array}$

Step by step solution

Concept of Differential equation

Differential equations are equations that connect one or more derivatives of a function. This implies that their answer is a function.

Step 2:Determining equation with the help of Leibniz’ rule

The equation is given as:

$\begin{array}{l} L_{n} (x) = \sum_{m = 0}^{n} {(- 1)}^{m} n_{c_{m}} \frac{x^{m}}{m!} \\ L_{n} (x) = \sum_{m = 0}^{n} {(- 1)}^{m} n_{c_{m}} \frac{x^{m}}{m!} \\ L_{0} (x) = 1 \\ L_{1} (x) = \sum_{m = 0}^{1} {(- 1)}^{m} 1_{c_{m}} \frac{x^{m}}{m!} \end{array}$

So, the value obtained of $L_{1} (x)$ is $1 - x$ .

$\begin{array}{l} L_{2} (x) = \sum_{m = 0}^{2} {(- 1)}^{m} 2_{c_{m}} \frac{x^{m}}{m!} \\ L_{2} (x) = 1 - 2 x + \frac{x^{2}}{2!} \\ L_{3} (x) = \sum_{m = 0}^{3} {(- 1)}^{m} 3_{c_{m}} \frac{x^{m}}{m!} \\ L_{3} (x) = 1 - x + \frac{3 x^{2}}{2} - \frac{x^{3}}{6} \end{array}$

Similarly find the series for $L_{4} (x)$ .

$L_{4} (x) = 1 - 4 x + 3 x^{2} - \frac{2 x^{3}}{3} + \frac{x^{4}}{24}$

The results for the given sequence are:

$\begin{array}{l} L_{3} (x) = 1 - x + \frac{3 x^{2}}{2} - \frac{x^{3}}{6} \\ L_{4} (x) = 1 - 4 x + 3 x^{2} - \frac{2 x^{3}}{3} + \frac{x^{4}}{24} \end{array}$ .

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Usingverify $(22 .19)$ and $(22.20)$ also find $L_{3} (x)$ and $L_{4} (x)$ .

Short Answer

Step by step solution

Concept of Differential equation

Step 2:Determining equation with the help of Leibniz’ rule

One App. One Place for Learning.

Most popular questions from this chapter

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Usingverify (22.19)and (22.20)also findL3(x)andL4(x).

Short Answer

Step by step solution

Concept of Differential equation

Step 2:Determining equation with the help of Leibniz’ rule

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Force

Electromagnetism

Engineering Physics

Linear Momentum

Torque and Rotational Motion

Geometrical and Physical Optics

Study anywhere. Anytime. Across all devices.

Usingverify $(22 .19)$ and $(22.20)$ also find $L_{3} (x)$ and $L_{4} (x)$ .