Why is this linear? d^3y/dt^3 + t * dy/dt + (cos(t))^2 * y = 4t^3

[bbqqcc123]

I was given this question to find out the order and decide whether it is linear or non-linear.
The answer provided is third-order and linear. I can understand why it is third-order but don't get why it is linear.
My understanding of a linear equation is shown below:
1) ...+y''+y'=y occur linearly.
2) Coefficient of y, y', y'' + ... must be the variable it correspond to.
for this question, it doesn't meet the first condition where y'' is missing.
Can someone help?

 

[khansaheb]

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View attachment 36593
I was given this question to find out the order and decide whether it is linear or non-linear.
The answer provided is third-order and linear. I can understand why it is third-order but don't get why it is linear.
My understanding of a linear equation is shown below:
1) ...+y''+y'=y occur linearly.
2) Coefficient of y, y', y'' + ... must be the variable it correspond to.
for this question, it doesn't meet the first condition where y'' is missing.
Can someone help?

Would you say:

A * y"' + B * y" + C * y = D

is a linear DE?

 

[topsquark]

View attachment 36592
View attachment 36593
I was given this question to find out the order and decide whether it is linear or non-linear.
The answer provided is third-order and linear. I can understand why it is third-order but don't get why it is linear.
My understanding of a linear equation is shown below:
1) ...+y''+y'=y occur linearly.
2) Coefficient of y, y', y'' + ... must be the variable it correspond to.
for this question, it doesn't meet the first condition where y'' is missing.
Can someone help?

What is the coefficient of y''? Is it a (non-linear) function of y?

-Dan

 

[Dr.Peterson]

I can understand why it is third-order but don't get why it is linear.
My understanding of a linear equation is shown below:
1) ...+y''+y'=y occur linearly.
2) Coefficient of y, y', y'' + ... must be the variable it correspond to.
for this question, it doesn't meet the first condition where y'' is missing.
Can someone help?

Please quote exactly the definition you were given. In your paraphrase, it is not at all clear what "occur linearly" and "must be the variable it correspond to" mean.

Here is the definition from Wikipedia:

In mathematics, a linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form​

[math]{\displaystyle a_{0}(x)y+a_{1}(x)y'+a_{2}(x)y''\cdots +a_{n}(x)y^{(n)}=b(x)}[/math]​

where a₀(x), ..., a_n(x) and b(x) are arbitrary differentiable functions that do not need to be linear, and y′, ..., y(n) are the successive derivatives of an unknown function y of the variable x.​

You seem to be supposing that none of the coefficients (functions of x) can be zero. That is not a requirement for linear polynomials.

 

[bbqqcc123]

Would you say:

A * y"' + B * y" + C * y = D

is a linear DE?

I will say yes provided that ABCD is all constant. Pls correct me if I'm wrong sry.

 

[bbqqcc123]

Please quote exactly the definition you were given. In your paraphrase, it is not at all clear what "occur linearly" and "must be the variable it correspond to" mean.

Here is the definition from Wikipedia:

In mathematics, a linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form​

[math]{\displaystyle a_{0}(x)y+a_{1}(x)y'+a_{2}(x)y''\cdots +a_{n}(x)y^{(n)}=b(x)}[/math]​

where a₀(x), ..., a_n(x) and b(x) are arbitrary differentiable functions that do not need to be linear, and y′, ..., y(n) are the successive derivatives of an unknown function y of the variable x.​

You seem to be supposing that none of the coefficients (functions of x) can be zero. That is not a requirement for linear polynomials.

I'm so sorry to give the wrong definition. To be really honest, I wasn't given a definition. It is only observed when I'm doing the question with provided answer.
Thank you so much for your clear explanation! you're right, I was supposing that none of the coefficients (functions of x) can be zero. I understand now.

 

[Dr.Peterson]

I'm so sorry to give the wrong definition. To be really honest, I wasn't given a definition. It is only observed when I'm doing the question with provided answer.
Thank you so much for your clear explanation! you're right, I was supposing that none of the coefficients (functions of x) can be zero. I understand now.

This may be the most important lesson: When you are asked a question like this, the first thing to do is to find a definition! First look in your own class materials or other near context (because definitions sometimes vary), and then in a reputable source like Wikipedia (which is not always safe for controversial issues, but is generally good, though sometimes too advanced, for math topics).

Never try to just guess from examples!