Chapter 4: Q38E (page 186)

Find a particular solution to the given higher-order equation.
$y^{4} - 5 y'' + 4 y = 10 cost - 20 sint$

Short Answer

Expert verified

The particular solution is $y_{p} (t) = cost - 2 sint .$

Step by step solution

Consider the particular solution for the given differential equation.

Given equation,

$y^{4} - 5 y'' + 4 y = 10 cost - 20 sint ...... (1)$

Consider the particular solution is,

$y_{p} (t) = Acost + Bsint . ..... (2)$

Take first, second, third and fourth derivatives of the above equation,

$\begin{matrix} y_{p}' (t) = - Asint + Bcost \\ y_{p}'' (t) = - Acost - Bsint \\ y_{p}''' (t) = Asint - Bcost \\ {y_{p}}^{4} (t) = Acost + Bsint \end{matrix}$

Substitute value of $y_{p} (t), y_{p}'' (t)$ and ${y_{p}}^{4} (t)$ in the equation (1),

$\begin{matrix} y^{4} - 5 y'' + 4 y = 10 cost - 20 sint \\ Acost + Bsint - 5 (- Acost - Bsint) + 4 (Acost + Bsint) = 10 cost - 20 sint \\ 10 Acost + 10 Bsint = 10 cost - 20 sint \end{matrix}$

Comparing the coefficients of the above equation;

$\begin{matrix} 10 A = 10 \Rightarrow A = 1 \\ 10 B = - 20 \Rightarrow B = - 2 \end{matrix}$

Final conclusion.

Therefore, the particular solution of the equation (1),

$\begin{matrix} y_{p} (t) = Acost + Bsint \\ y_{p} (t) = cost - 2 sint \end{matrix}$

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Find a particular solution to the given higher-order equation.
$y^{4} - 5 y'' + 4 y = 10 cost - 20 sint$

Short Answer

Step by step solution

Consider the particular solution for the given differential equation.

Final conclusion.

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Find a particular solution to the given higher-order equation.y4-5y''+4y=10cost-20sint

Short Answer

Step by step solution

Consider the particular solution for the given differential equation.

Final conclusion.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Probability and Statistics

Calculus

Geometry

Pure Maths

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

Find a particular solution to the given higher-order equation.
$y^{4} - 5 y'' + 4 y = 10 cost - 20 sint$