Chapter 8: Q7P (page 422)

Solve the following differential equations by the methods discussed above and compare computer solutions.
$(D^{2} - 5 D + 6) y = 0$

Short Answer

Expert verified

The solution is. $y = c_{1} e^{2 x} + c_{2} e^{3 x}$

Step by step solution

Given information

A differential equation is given as. $(D^{2} - 5 D + 6) y = 0$

Differential equation.

A differential equation of the form $(D - a) (D - b) y = 0, a \neq b$ has general solution

$y = c_{1} e^{a x} + c_{2} e^{b x} .$

Find the solution of given differential equation.

Supposethat. $D = \frac{d}{d x}$

Then

$\begin{matrix} D y = \frac{d y}{d x} \\ D y = y^{'} \\ D^{2} y = \frac{d}{d x} (\frac{d y}{d x}) \\ \frac{d^{2} y}{d x^{2}} = y^{''} \end{matrix}$

Given differential equation is. $(D^{2} - 5 D + 6) y = 0$

Auxiliary equation is-

$\begin{matrix} D^{2} - 5 D + 6 = 0 \\ (D - 2) (D - 3) = 0 \\ D = 2 \\ D = 3 \end{matrix}$

These are unequal roots.

$\Rightarrow D = 2, D = 3$

Differential equation becomes

$\Rightarrow (D - 2) (D - 3) y = 0$

These are separable equation with solution

$y = c_{1} e^{2 x} and y = c_{2} e^{3 x}$

The general solution will be givenby. $y = c_{1} e^{2 x} + c_{2} e^{3 x}$

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Solve the following differential equations by the methods discussed above and compare computer solutions.
$(D^{2} - 5 D + 6) y = 0$

Short Answer

Step by step solution

Given information

Differential equation.

Find the solution of given differential equation.

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Solve the following differential equations by the methods discussed above and compare computer solutions.(D2−5D+6)y=0

Short Answer

Step by step solution

Given information

Differential equation.

Find the solution of given differential equation.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Engineering Physics

Thermodynamics

Electricity and Magnetism

Famous Physicists

Scientific Method Physics

Conservation of Energy and Momentum

Study anywhere. Anytime. Across all devices.

Solve the following differential equations by the methods discussed above and compare computer solutions.
$(D^{2} - 5 D + 6) y = 0$