Chapter 1: Q11-6P (page 1)

Reduce the equation
$x^{2} \frac{d^{2} y}{d x^{2}} + 2 x \frac{d y}{d x} - 5 y = 0$
to a differential equation with constant coefficients in $\frac{d^{2} y}{d z^{2}}, \frac{d y}{d z}$ , and y by the change of variable $x = e^{x}$ . (See Chapter 8, Section 7d.)

Short Answer

Expert verified

Hence, the required differential equation is $\frac{d^{2} y}{d z^{2}} + \frac{d y}{d z} - 5 y = 0$ .

Step by step solution

Higher order derivatives

If any function $f (x)$ has its derivative with respect to x that is $f' (x)$ , then further differentiation of $f' (x)$ with respect to x results higher order derivative of $f (x)$ .

Differentiate the given equation

The given equation is:

$x^{2} \frac{d^{2} y}{d x^{2}} + 2 x \frac{d y}{d x} - 5 y = 0$

Now, we have $x = e^{z} \Rightarrow \frac{d x}{d z} = e^{z} \Rightarrow \frac{d z}{d x} = e^{- z}$ . Then,

$\frac{d y}{d x} = \frac{d y}{d z} \times \frac{d z}{d x} = e^{- z} \frac{d y}{d z}$

And,

$\frac{d^{2} y}{d x^{2}} = \frac{d}{d x} (e^{- z} \frac{d y}{d z}) = \frac{d}{d z} (e^{- z} \frac{d y}{d z}) \frac{d z}{d x} = (e^{- z} \frac{d^{2} y}{d z^{2}} - e^{- z} \frac{d y}{d z}) e^{- z} = e^{- 2 z} (\frac{d^{2} y}{d z^{2}} - \frac{d y}{d z}) x^{2} \frac{d^{2} y}{d x^{2}} = \frac{d^{2} y}{d z^{2}} - \frac{d y}{d z}$

Solve the obtained derivatives

Substitute all these values in $x^{2} \frac{d^{2} y}{d x^{2}} + 2 x \frac{d y}{d x} - 5 y = 0$ , and we get:

$x^{2} \frac{d^{2} y}{d x^{2}} + 2 x \frac{d y}{d x} - 5 y = 0 (\frac{d^{2} y}{d z^{2}} - \frac{d y}{d z}) + 2 \frac{d y}{d x} - 5 y = 0 \frac{d^{2} y}{d z^{2}} + \frac{d y}{d z} - 5 y = 0$

Hence, the required differential equation is $\frac{d^{2} y}{d z^{2}} + \frac{d y}{d z} - 5 y = 0$ .

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Reduce the equation
$x^{2} \frac{d^{2} y}{d x^{2}} + 2 x \frac{d y}{d x} - 5 y = 0$
to a differential equation with constant coefficients in $\frac{d^{2} y}{d z^{2}}, \frac{d y}{d z}$ , and y by the change of variable $x = e^{x}$ . (See Chapter 8, Section 7d.)

Short Answer

Step by step solution

Higher order derivatives

Differentiate the given equation

Solve the obtained derivatives

One App. One Place for Learning.

Most popular questions from this chapter

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Reduce the equationx2d2ydx2+2xdydx-5y=0to a differential equation with constant coefficients in d2ydz2,dydz, and y by the change of variable x=ex. (See Chapter 8, Section 7d.)

Short Answer

Step by step solution

Higher order derivatives

Differentiate the given equation

Solve the obtained derivatives

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Energy Physics

Thermodynamics

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Further Mechanics and Thermal Physics

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Study anywhere. Anytime. Across all devices.

Reduce the equation
$x^{2} \frac{d^{2} y}{d x^{2}} + 2 x \frac{d y}{d x} - 5 y = 0$
to a differential equation with constant coefficients in $\frac{d^{2} y}{d z^{2}}, \frac{d y}{d z}$ , and y by the change of variable $x = e^{x}$ . (See Chapter 8, Section 7d.)