Chapter 1: Q21 E (page 1)

Determine which values of m the function $ϕ (x) = x^{m}$ is a solution to the given equation.
(a) $3 x^{2} \frac{d^{2} y}{{dx}^{2}} + 11 x \frac{dy}{dx} - 3 y = 0$
(b) $x^{2} \frac{d^{2} y}{{dx}^{2}} - x \frac{dy}{dx} - 5 y = 0$

Short Answer

Expert verified

$m = \frac{1}{3}, - 3$
$m = 1 \pm \sqrt{6}$

Step by step solution

(a): Taking the given function as y

First of all, we will take $ϕ (x) = y$

Differentiating the given function

Differentiatingconcerning x,

$ϕ' (x) = \frac{dy}{dx} = m x^{m - 1}$

Again, differentiating concerning x,

$ϕ'' (x) = \frac{d^{2} y}{{dx}^{2}} = m (m - 1) x^{m - 2}$

Substituting the values from step 2 in the given differential equation

$\begin{array}{l} 3 x^{2} \frac{d^{2} y}{{dx}^{2}} + 11 x \frac{dy}{dx} - 3 y = 0 \\ 3 x^{2} [m (m - 1) x^{m - 2}] + 11 (x) {mx}^{m - 1} - 3 x^{m} = 0 \\ 3 m^{2} x^{m} - 3 {mx}^{m} + 11 {mx}^{m} - 3 x^{m} = 0 \\ 3 m^{2} + 8 m - 3 = 0 \\ (m - \frac{1}{3}) (m + 3) = 0 \\ m = \frac{1}{3}, - 3 \end{array}$

Hence, the values of m are $\frac{1}{3}$ and -3.

Step 4(b): Taking the given function as y

First of all, we will take $ϕ (x) = y$ .

Differentiating the given function

Differentiatingconcerning x,

$ϕ' (x) = \frac{dy}{dx} = m x^{m - 1}$

Again, differentiating concerning x,

$ϕ'' (x) = \frac{d^{2} y}{{dx}^{2}} = m (m - 1) x^{m - 2}$

Substituting the values from step 2 in the given differential equation.

$\begin{array}{l} x^{2} \frac{d^{2} y}{{dx}^{2}} - x \frac{dy}{dx} - 5 y = 0 \\ x^{2} m (m - 1) x^{m - 2} - {xmx}^{m - 1} - 5 x^{m} = 0 \\ m^{2} x^{m} - {mx}^{m} - {mx}^{m} - 5 x^{m} = 0 \\ m^{2} x^{m} - 2 {mx}^{m} - 5 x^{m} = 0 \\ (m^{2} - 2 m - 5) x^{m} = 0 \\ m^{2} - 2 m - 5 = 0 \\ m = 1 \pm \sqrt{6} \end{array}$

Hence, the values of m are $(1 + \sqrt{6})$ and $(1 - \sqrt{6})$ .

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Determine which values of m the function $ϕ (x) = x^{m}$ is a solution to the given equation.
(a) $3 x^{2} \frac{d^{2} y}{{dx}^{2}} + 11 x \frac{dy}{dx} - 3 y = 0$
(b) $x^{2} \frac{d^{2} y}{{dx}^{2}} - x \frac{dy}{dx} - 5 y = 0$

Short Answer

Step by step solution

(a): Taking the given function as y

Differentiating the given function

Substituting the values from step 2 in the given differential equation

Step 4(b): Taking the given function as y

Differentiating the given function

Substituting the values from step 2 in the given differential equation.

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Determine which values of m the functionϕ(x)=xmis a solution to the given equation.(a)3x2d2ydx2+11xdydx-3y=0(b)x2d2ydx2-xdydx-5y=0

Short Answer

Step by step solution

(a): Taking the given function as y

Differentiating the given function

Substituting the values from step 2 in the given differential equation

Step 4(b): Taking the given function as y

Differentiating the given function

Substituting the values from step 2 in the given differential equation.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

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Decision Maths

Study anywhere. Anytime. Across all devices.

Determine which values of m the function $ϕ (x) = x^{m}$ is a solution to the given equation.
(a) $3 x^{2} \frac{d^{2} y}{{dx}^{2}} + 11 x \frac{dy}{dx} - 3 y = 0$
(b) $x^{2} \frac{d^{2} y}{{dx}^{2}} - x \frac{dy}{dx} - 5 y = 0$