Chapter 4: Q9E (page 164)

Question: find a general solution to the given differential equation.
$4 y'' - 4 y' + y = 0$

Short Answer

Expert verified

Answer

The general solution of the given equation is $y = (c_{1} + c_{2} t) e^{\frac{1}{2} t} .$

Step by step solution

Write the auxiliary equation of the given differential equation.

The given differential equation is

$4 y'' - 4 y' + y = 0 .$

The auxiliary equation for the above equation

$4 m^{2} - 4 m + 1 = 0 .$

Now find the roots of the auxiliary equation.

Solve the auxiliary equation,

$4 m^{2} - 4 m + 1 = 0 {(2 m)}^{2} - 2 (2) (1) m + 1 = 0 {(2 m - 1)}^{2} = 0$

The roots of the auxiliary equation are $m_{1} = \frac{1}{2}, & m_{2} = \frac{1}{2} .$

Write the general solution.

If the auxiliary equation has repeated real roots, then the general solution is given as;

$y = (c_{1} + c_{2} t) e^{mt}$

Thus, the general solution of the given equation is $y = (c_{1} + c_{2} t) e^{\frac{1}{2} t} .$

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Question: find a general solution to the given differential equation.
$4 y'' - 4 y' + y = 0$

Short Answer

Step by step solution

Write the auxiliary equation of the given differential equation.

Now find the roots of the auxiliary equation.

Write the general solution.

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Question: find a general solution to the given differential equation.4y''-4y'+y=0

Short Answer

Step by step solution

Write the auxiliary equation of the given differential equation.

Now find the roots of the auxiliary equation.

Write the general solution.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Decision Maths

Geometry

Pure Maths

Applied Mathematics

Statistics

Study anywhere. Anytime. Across all devices.

Question: find a general solution to the given differential equation.
$4 y'' - 4 y' + y = 0$