Chapter 4: Q9E (page 180)

Find a particular solution to the differential equation. $y'' + 3 y = - 9$

Short Answer

Expert verified

Thus, the particular solution to the differential equation is $y_{p} (x) = - 3 .$

Step by step solution

Firstly, write the auxiliary equation of the given differential equation.

The given differential equation,

$y'' + 3 y = - 9 \dots (1)$

The auxiliary equation for the above equation,

$m^{2} + 3 = 0$

Now find the roots of the auxiliary equation.

Solve the auxiliary equation,

$\begin{matrix} m^{2} + 3 = 0 \\ m^{2} = - 3 \\ m = \pm \sqrt{- 3} \\ m = \pm i \sqrt{3} \end{matrix}$

The roots of the auxiliary equation are:

$m_{1} = i \sqrt{3} & m_{2} = - i \sqrt{3}$

The complementary solution of the given equation is;

$\begin{matrix} y_{c} (x) = e^{0 \cdot t} [c_{1} \cos (\sqrt{3} t) + c_{2} \sin (\sqrt{3} t)] \\ = c_{1} \cos (\sqrt{3} t) + c_{2} \sin (\sqrt{3} t) \end{matrix}$

Final conclusion, find a particular solution to the differential equation.

Assume, the particular solution of equation (1),

$y_{p} (x) = A \dots (2)$

Now find the derivative of the above equation,

$\begin{matrix} y_{p}' (x) = 0 \\ y_{p}'' (x) = 0 \end{matrix}$

From the equation (1),

$\begin{matrix} y_{p}'' + 3 y_{p} = - 9 \\ (0) + 3 A = - 9 \\ A = - 3 \end{matrix}$

Substitute the value of A in the equation (2), and we get:

$y_{p} (x) = - 3$

Therefore, the particular solution of the given differential equation is:

$y_{p} (x) = - 3$

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.

Find a particular solution to the differential equation. $y'' + 3 y = - 9$

Short Answer

Step by step solution

Firstly, write the auxiliary equation of the given differential equation.

Now find the roots of the auxiliary equation.

Final conclusion, find a particular solution to the differential equation.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Geometry

Decision Maths

Discrete Mathematics

Statistics

Pure Maths

Study anywhere. Anytime. Across all devices.

Find a particular solution to the differential equation.y''+3y=-9

Short Answer

Step by step solution

Firstly, write the auxiliary equation of the given differential equation.

Now find the roots of the auxiliary equation.

Final conclusion, find a particular solution to the differential equation.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Geometry

Decision Maths

Discrete Mathematics

Statistics

Pure Maths

Study anywhere. Anytime. Across all devices.

Find a particular solution to the differential equation. $y'' + 3 y = - 9$