Chapter 4: Q40E (page 186)

Find a particular solution to the given higher-order equation. $y^{4} - 3 y''' + 3 y'' - y' = 6 t - 20$

Short Answer

Expert verified

The particular solution to the given higher-order equation is $y_{p} (t) = - 3 t^{2} + 2 t .$

Step by step solution

Consider the particular solution for the given differential equation.

The given differential equation is,

$y^{4} - 3 y''' + 3 y'' - y' = 6 t - 20 . ..... (1)$

Consider the particular solution is,

$y_{p} (t) = {At}^{2} + Bt . .... (2)$

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

$\begin{matrix} y_{p}' (t) = 2 At + B \\ y_{p}'' (t) = 2 A \\ y_{p}''' (t) = 0 \\ {y_{p}}^{4} (t) = 0 \end{matrix}$

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

$\begin{matrix} y^{4} - 3 y''' + 3 y'' - y' = 6 t - 20 \\ 0 - 3 (0) + 3 (2 A) - (2 At + B) = 6 t - 20 \\ - 2 At + (- B + 6 A) = 6 t - 20 \end{matrix}$

Comparing all coefficients of the above equation;

$\begin{matrix} - 2 A = 6 \Rightarrow A = - 3 \\ - B + 6 A = - 20 . ..... (3) \end{matrix}$

Substitute the value A in the equation (3),

$\begin{matrix} - B + 6 (- 3) = - 20 \\ B = 2 \end{matrix}$

Final conclusion.

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

$\begin{array}{l} y_{p} (t) = {At}^{2} + Bt \\ y_{p} (t) = - 3 t^{2} + 2 t \end{array}$

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Find a particular solution to the given higher-order equation. $y^{4} - 3 y''' + 3 y'' - y' = 6 t - 20$

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.y4-3y'''+3y''-y'=6t-20

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

Geometry

Discrete Mathematics

Decision Maths

Logic and Functions

Theoretical and Mathematical Physics

Pure Maths

Study anywhere. Anytime. Across all devices.

Find a particular solution to the given higher-order equation. $y^{4} - 3 y''' + 3 y'' - y' = 6 t - 20$