# Linearity in differential equations

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1. Feb 21, 2015

### Chris B

1. The problem statement, all variables and given/known data
Is the equation
(x2sinx + 4y) dx + x dy=0
linear
This problem also asks me to solve it, but I don't have a problem with that part.

2. Relevant equations
An equation is linear if the function or its derivative are only raised to the first power and not multiplied by each other.

3. The attempt at a solution
The answer in the back of the book says that it's linear with x as the dependent variable. I tried rearranging so that all x's were to the first power, but nothing doing. Is it a typo and it meant it's linear with y as the dependent variable? Because y is already by itself, and if you divide by dx then y' and y aren't multiplied by each other either so
x2sinx + 4y =-x dy/dx

Right? Have I mixed up dependent and independent somehow?

2. Feb 21, 2015

### vela

Staff Emeritus
"The function" refers to y.

3. Feb 21, 2015

### Chris B

Fair enough, but if y is the function then x is the independent variable, right?

4. Feb 21, 2015

### AfterSunShine

Right.

5. Feb 21, 2015

### Chris B

Okay, then my book's answer key is wrong. Thanks.

6. Feb 21, 2015

### AfterSunShine

You are welcome.
" An equation is linear if the function or its derivative are only raised to the first power, not multiplied by each other and not composited with other function ."

i.e. there is no cos(y), ln(y) , arctan(y'), e^y .. etc