Short Answer

In Problems 3–8, determine whether the given function is a solution to the given differential equation.
$θ = 2 e^{3 t} - e^{2 t}$ , $\frac{d^{2} θ}{{dt}^{2}} - θ \frac{dθ}{dt} + 3 θ = - 2 e^{2 t}$

The given function is not a solution to the given differential equation.

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Differentiating the given equation w.r.t. (with respect to) t

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

$\frac{dθ}{dt} = 6 e^{3 t} - 2 e^{2 t}$

Again, differentiating with respect to t,

$\frac{d^{2} θ}{{dt}^{2}} = 18 e^{3 t} - 4 e^{2 t}$

Step 2: Simplification

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

$\frac{d^{2} θ}{{dt}^{2}} - θ \frac{d θ}{dt} + 3 θ = 18 e^{3 t} - 4 e^{2 t} - (2 e^{3 t} - e^{2 t}) \times (6 e^{3 t} - 2 e^{2 t}) + 3 \times (2 e^{3 t} - e^{2 t}) \frac{d^{2} θ}{{dt}^{2}} - θ \frac{d θ}{dt} + 3 θ = 18 e^{3 t} - 4 e^{2 t} - (12 e^{6 t} - 4 e^{5 t} - 6 e^{5 t} + 2 e^{4 t}) + 6 e^{3 t} - 3 e^{2 t} \frac{d^{2} θ}{{dt}^{2}} - θ \frac{dθ}{dt} + 3 θ = 24 e^{3 t} - 7 e^{2 t} - (12 e^{6 t} - 4 e^{5 t} - 6 e^{5 t} + 2 e^{4 t})$

which is not the same as the R.H.S. (Right-hand side) of the given differential equation.

Hence, $θ = 2 e^{3 t} - e^{2 t}$ is not a solution to the differential equation $\frac{d^{2} θ}{{dt}^{2}} - θ \frac{dθ}{dt} + 3 θ = - 2 e^{2 t}$ .

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Fundamentals Of Differential Equations And Boundary Value Problems

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Short Answer

In Problems 3–8, determine whether the given function is a solution to the given differential equation.
$θ = 2 e^{3 t} - e^{2 t}$ , $\frac{d^{2} θ}{{dt}^{2}} - θ \frac{dθ}{dt} + 3 θ = - 2 e^{2 t}$

Step by Step Solution

Step 1: Differentiating the given equation w.r.t. (with respect to) t

Step 2: Simplification

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Fundamentals Of Differential Equations And Boundary Value Problems

Answers without the blur.

Short Answer

In Problems 3–8, determine whether the given function is a solution to the given differential equation.θ=2e3t-e2t, d2θdt2-θdθdt+3θ=-2e2t

Step by Step Solution

Step 1: Differentiating the given equation w.r.t. (with respect to) t

Step 2: Simplification

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In Problems 3–8, determine whether the given function is a solution to the given differential equation.
$θ = 2 e^{3 t} - e^{2 t}$ , $\frac{d^{2} θ}{{dt}^{2}} - θ \frac{dθ}{dt} + 3 θ = - 2 e^{2 t}$