The second-order homogeneous linear differential equation whose characteristic equation has roots
−
3
8
+
1
4
𝑖
and
−
3
8
−
1
4
𝑖
is:

Hint: A quadratic equation whose roots are
𝛼
and
𝛽
is:
𝑋
2
−
(
𝛼
+
𝛽
)
𝑋
+
𝛼
𝛽
=
0
.

Question

The second-order homogeneous linear differential equation whose characteristic equation has roots 
−
3
8
+
1
4
𝑖
 and 
−
3
8
−
1
4
𝑖
 is:

Hint: A quadratic equation whose roots are 
𝛼
 and 
𝛽
 is:   
𝑋
2
−
(
𝛼
+
𝛽
)
𝑋
+
𝛼
𝛽
=
0
.

BlackTom AI · Accepted Answer

The problem asks for the second-order homogeneous linear differential equation whose characteristic roots are -3/8 ± i/4. 
First, recall that if the roots of the characteristic equation are α and β, the corresponding quadratic is X^2 − (α + β) X + αβ = 0. Here, the two roots are complex conjugates: α = -3/8 + i/4 and β = -3/8 − i/4.
Compute the sum: α + β = (-3/8 + i/4) + (-3/8 − i/4) = -3/4. 
Compute the product: αβ = (-3/8 + i/4)(-3/8 − i/4) = (-3/8)^2 + (1/4)^2 = 9/64 + 1/16 = 9/64 + 4/64 = 13/64.
Thus the quadratic with these roots is X^2 − (−3/4) X + 13/64 = X^2 + (3/4) X + 13/64 = 0.
To convert to a standard differential equation form, use the fact that if the characteristic equation is r^2 + p r + q = 0, then the corresponding differential equation is y'' + p y' + q y = 0. Here, p = 3/4 and q = 13/64.
Clear denominators by multiplying through by 64 to obtain integer coefficients: 64 y'' + 48 y' + 13 y = 0.
Therefore, the differential equation that has the given roots is 64 x'' + 48 x' + 13 x = 0. 
A quick consistency check: the derived p and q match the real and imaginary parts of the roots when forming the quadratic, and the multiplication by 64 yields clean integer coefficients as shown, which aligns with typical expectations for such problems.

SMAT011 Weekly Quiz 9 |LA009

The second-order homogeneous linear differential equation whose characteristic equation has roots − 3 8 + 1 4 𝑖 and − 3 8 − 1 4 𝑖 is: Hint: A quadratic equation whose roots are 𝛼 and 𝛽 is: 𝑋 2 − ( 𝛼 + 𝛽 ) 𝑋 + 𝛼 𝛽 = 0 .

查看解析

登录即可查看完整答案

类似问题

In a consumer society, many adults channel creativity into buying things

Economic stress and unpredictable times have resulted in a booming industry for self-help products

People born without creativity never can develop it

A product has a selling price of $20, a contribution margin ratio of 40% and fixed cost of $120,000. To make a profit of $30,000. The number of units that must be sold is: Type the number without $ and a comma. Eg: 20000

Which of the following statement regarding cost is correct:

Under the assumptions used in cost-volume-profit analysis, as the activity increases:

The CVP and break-even analysis are supported by a range of important assumptions. Which of the following is not an assumption of the CVP:

更多留学生实用工具

考试浏览器助手

风格化写作助手

论文查重助手

文献引用助手

课堂转译助手

课堂笔记助手

Quiz搜索助手

学校历年真题

智能刷题助手

智能匹配练习

希望你的学习变得更简单