Consider the first order linear differential equation x
dy

dx
=3y−4x. What is the integrating factor r(x)?

Question

Consider the first order linear differential equation x
dy

dx
=3y−4x. What is the integrating factor r(x)?

BlackTom AI · Accepted Answer

To begin, transform the given first-order linear differential equation into the standard form. The equation x dy/dx = 3y - 4x can be rearranged by dividing both sides by x (assuming x ≠ 0): dy/dx = (3/x) y - 4, which is the same as dy/dx - (3/x) y = -4. In this standard form, the integrating factor μ(x) is exp(∫ P(x) dx) where P(x) is the coefficient of y, here P(x) = -3/x.

Compute the integrating factor: ∫(-3/x) dx = -3 ln|x|, so μ(x) = exp(-3 ln|x|) = |x|^{-3}. For simplicity and customary use in many problems, this is written as x^{-3} (valid on intervals where x keeps a constant sign).

Now compare this result to the given options. Option 1: x^{-3} matches the expression derived for the integrating factor (up to the convention about the sign of x, which is handled by domain considerations).

Option 2: e^{-3x} would correspond to integrating -3 as the coefficient of dx, which is not the case here because the P(x) term is -3/x, not a constant times x. Thus e^{-3x} is incorrect for this integrating factor.

Option 3: x^4 would imply an integrating factor that grows as a power of x^4, which does not align with μ(x) = x^{-3} derived from -3/x. Therefore this is incorrect.

Option 4: e^{4x} similarly represents an exponential integrating factor in x, inconsistent with the -3/x dependence in P(x). Hence this option is not correct.

In summary, the integrating factor for the equation, considering the standard method and the coefficient of y, is a power of x with exponent -3, i.e., x^{-3}. The comparison shows that only the first option aligns with this result.

MATH1062/1005/1023 (ND) MATH1062/1023 Calculus Quiz 4

Consider the first order linear differential equation x dy dx =3y−4x. What is the integrating factor r(x)?

查看解析

登录即可查看完整答案

类似问题

What is the purpose of using an integrating factor when solving a first-order linear differential equation?

Which of the following five differential equations cannot be solved using an integrating factor? (More than one correct answer.)

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:

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