# Question on Linearity

1. Oct 19, 2014

### RiceKernel

Hi , I have no problem to solve but just a bit of confusion on what determines the linearity of an ODE.

Let's say the equation is (1 x^2) dy/dx + y = 0

Is it linear ? I would incline to say yes because the dependent variable and its derivatives are not in a product with each other but the square on the x makes me doubt the linearity or does it not matter at all? If it was (1-y^2), it wouldn't be linear because the coefficient has the dependent variable in it.

Thanks ,
GT

2. Oct 19, 2014

### hilbert2

An ODE is linear if the sum of any two solutions to it is also a solution.

Suppose we have two functions, $y_{1}$ and $y_{2}$, and they both satisfy the ODE:

$(1-x^{2})\frac{dy_{1}}{dx} + y_{1}=0$
$(1-x^{2})\frac{dy_{2}}{dx} + y_{2}=0$

Can you show that $y_{1}+y_{2}$ also satisfies the ODE? If you can, then the eq is linear.

3. Oct 20, 2014

### RiceKernel

Not helpful at all . Just had to say yes or no .

4. Oct 20, 2014

### bigfooted

give a man a fish...

5. Dec 14, 2014