Chapter 1: Q7 E (page 14)

In Problems 3–8, determine whether the given function is a solution to the given differential equation.
$y = e^{2 x} - 3 e^{- x}$ , $\frac{d^{2} y}{{dx}^{2}} - \frac{dy}{dx} - 2 y = 0$

Short Answer

Expert verified

The given function is a solution to the given differential equation.

Step by step solution

Differentiating the given equation w.r.t. (with respect to) x

Firstly, we will differentiate $y = e^{2 x} - 3 e^{- x}$ with respect to x,

$\frac{dy}{dx} = 2 e^{2 x} + 3 e^{- x}$

Again, differentiating the given function with respect to x,

$\frac{d^{2} y}{{dx}^{2}} = 4 e^{2 x} - 3 e^{- x}$

Simplification

Putting the values from step 1 in the L.H.S. (Left-hand side) of the given differential equation,

$\frac{d^{2} y}{{dx}^{2}} - \frac{dy}{dx} - 2 y = 4 e^{2 x} - 3 e^{- x} - (2 e^{2 x} + 3 e^{- x}) - 2 (e^{2 x} - 3 e^{- x}) \frac{d^{2} y}{{dx}^{2}} - \frac{dy}{dx} - 2 y = 4 e^{2 x} - 3 e^{- x} - 2 e^{2 x} - 3 e^{- x} - 2 e^{2 x} + 6 e^{- x} \frac{d^{2} y}{{dx}^{2}} - \frac{dy}{dx} - 2 y = 0$

which is the same as the R.HS. (Right-hand side) of the given differential equation.

Hence, $y = e^{2 x} - 3 e^{- x}$ is a solution to the differential equation $\frac{d^{2} y}{{dx}^{2}} - \frac{dy}{dx} - 2 y = 0$ .

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In Problems 3–8, determine whether the given function is a solution to the given differential equation.
$y = e^{2 x} - 3 e^{- x}$ , $\frac{d^{2} y}{{dx}^{2}} - \frac{dy}{dx} - 2 y = 0$

Short Answer

Step by step solution

Differentiating the given equation w.r.t. (with respect to) x

Simplification

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In Problems 3–8, determine whether the given function is a solution to the given differential equation.y=e2x-3e-x,d2ydx2-dydx-2y=0

Short Answer

Step by step solution

Differentiating the given equation w.r.t. (with respect to) x

Simplification

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Applied Mathematics

Geometry

Decision Maths

Logic and Functions

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

In Problems 3–8, determine whether the given function is a solution to the given differential equation.
$y = e^{2 x} - 3 e^{- x}$ , $\frac{d^{2} y}{{dx}^{2}} - \frac{dy}{dx} - 2 y = 0$