Chapter 1: Q5 E (page 13)

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

Short Answer

Expert verified

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

Step by step solution

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, differentiatingwith respect to t,

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

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 $\frac{d^{2} θ}{{dt}^{2}} - θ \frac{dθ}{dt} + 3 θ = - 2 e^{2 t}$ .

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

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

Short Answer

Step by step solution

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

Simplification

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Logic and Functions

Probability and Statistics

Decision Maths

Calculus

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

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

Short Answer

Step by step solution

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

Simplification

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Logic and Functions

Probability and Statistics

Decision Maths

Calculus

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

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