Chapter 6: Q21E (page 332)

Solve the given initial value problem
_{$\begin{array}{l} y''' - 4 y'' + 7 y' - 6 y = 0 \\ y (0) = 1 \\ y' (0) = 0 \\ y'' (0) = 0 \end{array}$}

Short Answer

Expert verified

_{The solution is}_{$y (t) = e^{2 t} - \sqrt{2} e^{t} \sin \sqrt{2} t$}

Step by step solution

Basic differentiation

The Sum rule says the derivative of a sum of functions is the sum of their derivatives. The Difference rule says the derivative of a difference of functions is the difference of their derivatives.

Solving by basic differentiation:

We will do the following question on the basis of basic differentiation

$\begin{array}{l} r^{3} - 4 r^{2} + 7 r - 6 = 0 \\ r^{3} - 4 r^{2} + 7 r - 6 = (r - 2) (r^{2} - 2 r + 3) = 0 \\ y (t) = c_{1} e^{2 t} + c_{2} e^{t} \sin \sqrt{2} t + c_{3} e^{t} \cos \sqrt{2} t \\ y' (t) = 2 c_{1} e^{2 t} + c_{2} e^{t} \sin \sqrt{2} t + \sqrt{2} c_{2} e^{t} \cos \sqrt{2} t + c_{3} e^{t} \cos \sqrt{2} t - \sqrt{2} c_{3} e^{t} \sin \sqrt{2} t \\ y'' (t) = 4 c_{1} e^{2 t} + c_{2} e^{t} \sin \sqrt{2} t + \sqrt{2} c_{2} e^{t} \cos \sqrt{2} t + \sqrt{2} c_{2} e^{t} \cos \sqrt{2} t - \sqrt{2} c_{2} e^{t} \sin \sqrt{2} t + c_{3} e^{t} \cos \sqrt{2} t - \sqrt{2} c_{3} e^{t} \sin \sqrt{2} t - \sqrt{2} c_{3} e^{t} \sin \sqrt{2} t - \sqrt{2} c_{3} e^{t} \cos \sqrt{2} t \\ y (0) = 1 \\ y' (0) = 0 \\ y'' (0) = 0 \\ y (t) = e^{2 t} - \sqrt{2} e^{t} \sin \sqrt{2} t \end{array}$

Hence, the final answer is:

$y (t) = e^{2 t} - \sqrt{2} e^{t} \sin \sqrt{2} t$

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Solve the given initial value problem
_{$\begin{array}{l} y''' - 4 y'' + 7 y' - 6 y = 0 \\ y (0) = 1 \\ y' (0) = 0 \\ y'' (0) = 0 \end{array}$}

Short Answer

Step by step solution

Basic differentiation

Solving by basic differentiation:

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Solve the given initial value problem y'''−4y''+7y'−6y=0y(0)=1y'(0)=0y''(0)=0

Short Answer

Step by step solution

Basic differentiation

Solving by basic differentiation:

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Decision Maths

Applied Mathematics

Mechanics Maths

Discrete Mathematics

Calculus

Study anywhere. Anytime. Across all devices.

Solve the given initial value problem
_{$\begin{array}{l} y''' - 4 y'' + 7 y' - 6 y = 0 \\ y (0) = 1 \\ y' (0) = 0 \\ y'' (0) = 0 \end{array}$}