Chapter 1: Q20 E (page 14)

Determine which values of m the function $ϕ (x) = e^{mx}$ is a solution to the given equation.
(a) $\frac{d^{2} y}{{dx}^{2}} + 6 \frac{dy}{dx} + 5 y = 0$
(b) $\frac{d^{3} y}{{dx}^{3}} + 3 \frac{d^{2} y}{{dx}^{2}} + 2 \frac{dy}{dx} = 0$

Short Answer

Expert verified

$m = - 1, - 5$
$m = - 1, - 2$

Step by step solution

Step 1(a): Taking the given function as y

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

Differentiating the given function

Differentiating $ϕ (x) = e^{mx}$ with respect to x,

$ϕ' (x) = \frac{dy}{dx} = {me}^{mx}$

Again, differentiating with respect to x,

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

Substituting the values from step 2 in the given differential equation

$\frac{d^{2} y}{{dx}^{2}} + 6 \frac{dy}{dx} + 5 y = 0 m^{2} e^{mx} + 6 {me}^{mx} + 5 e^{mx} = 0 (m^{2} + 6 m + 5) e^{mx} = 0 m^{2} + 6 m + 5 = 0 (m + 1) (m + 5) = 0 m = - 1, - 5$

Hence, the values of m are -1 and -5.

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

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

Differentiating the given function

Differentiating $ϕ (x) = e^{mx}$ with respect to x,

$ϕ' (x) = \frac{dy}{dx} = {me}^{mx}$

Again, differentiating with respect to x,

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

Again, differentiating with respect to x,

$ϕ''' (x) = \frac{d^{3} y}{{dx}^{3}} = m^{3} e^{mx}$

Substituting the values from step 2 in the given differential equation

$\begin{array}{l} \frac{d^{3} y}{{dx}^{3}} + 3 \frac{d^{2} y}{{dx}^{2}} + 2 \frac{dy}{dx} = 0 \\ m^{3} e^{mx} + 3 m^{2} e^{mx} + 2 {me}^{mx} = 0 \\ {me}^{mx} (m^{2} + 3 m + 2) = 0 \\ m^{2} + 3 m + 2 = 0 \\ (m + 1) (m + 2) = 0 \\ m = - 1, - 2 \end{array}$

Hence, the values of m are -1 and -2.

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Determine which values of m the function $ϕ (x) = e^{mx}$ is a solution to the given equation.
(a) $\frac{d^{2} y}{{dx}^{2}} + 6 \frac{dy}{dx} + 5 y = 0$
(b) $\frac{d^{3} y}{{dx}^{3}} + 3 \frac{d^{2} y}{{dx}^{2}} + 2 \frac{dy}{dx} = 0$

Short Answer

Step by step solution

Step 1(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 function ϕ(x)=emxis a solution to the given equation.(a)d2ydx2+6dydx+5y=0(b)d3ydx3+3d2ydx2+2dydx=0

Short Answer

Step by step solution

Step 1(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 function $ϕ (x) = e^{mx}$ is a solution to the given equation.
(a) $\frac{d^{2} y}{{dx}^{2}} + 6 \frac{dy}{dx} + 5 y = 0$
(b) $\frac{d^{3} y}{{dx}^{3}} + 3 \frac{d^{2} y}{{dx}^{2}} + 2 \frac{dy}{dx} = 0$