Chapter 4: Q4E (page 180)

Decide whether or not the method of undetermined coefficients can be applied to find a particular solution of the given equation. $x'' + 5 x' - 3 x = 3^{t}$

Short Answer

Expert verified

Yes, the method of undetermined coefficients can be applied.

Step by step solution

Use logarithms properties for simplification of the given differential equation.

Given equation,

$x'' + 5 x' - 3 x = 3^{t}$

Simplify the above equation by using logarithms properties,

$\begin{matrix} x'' + 5 x' - 3 x = e^{\ln (3^{t})} \\ x'' + 5 x' - 3 x = e^{t [\ln (3)]} \\ x'' + 5 x' - 3 x = e^{[\ln (3)] t} \end{matrix}$

Use the method of undetermined coefficients to find a particular solution of a given differential equation.

The given differential equation is in the form of;

$ax'' + bx' + cx = e^{rt}$

According to the method of undetermined coefficients,

To find a particular solution to the differential equation;

$ay'' (x) + by' (x) + cy (x) = {Ct}^{m} e^{rt}$

Where m is a non-negative integer, use the form;

$y_{p} (x) = t^{s} (A_{m} t^{m} + . .. + A_{1} t + A_{0}) e^{rt}$

Compare with the given differential equation,

$x'' + 5 x' - 3 x = e^{[\ln (3)] t}$

Condition satisfies,

s = 1 if r is a simple root of the associated auxiliary equation.

Therefore, the particular solution of the equation,

$y_{p} (x) = {Ate}^{\ln (3) t}$

So, the method of undetermined coefficients can be applied.

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Decide whether or not the method of undetermined coefficients can be applied to find a particular solution of the given equation. $x'' + 5 x' - 3 x = 3^{t}$

Short Answer

Step by step solution

Use logarithms properties for simplification of the given differential equation.

Use the method of undetermined coefficients to find a particular solution of a given differential equation.

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Decide whether or not the method of undetermined coefficients can be applied to find a particular solution of the given equation. x''+5x'-3x=3t

Short Answer

Step by step solution

Use logarithms properties for simplification of the given differential equation.

Use the method of undetermined coefficients to find a particular solution of a given differential equation.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Applied Mathematics

Geometry

Decision Maths

Mechanics Maths

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

Decide whether or not the method of undetermined coefficients can be applied to find a particular solution of the given equation. $x'' + 5 x' - 3 x = 3^{t}$