Linearity of ODE: (1 x^2) dy/dx + y = 0

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SUMMARY

The discussion centers on the linearity of the ordinary differential equation (ODE) represented by (1 x^2) dy/dx + y = 0. The equation is confirmed to be linear because the dependent variable y and its derivative dy/dx do not multiply each other, despite the presence of the x^2 term. A key criterion for linearity is that the sum of any two solutions must also be a solution, which holds true for this ODE. The confusion arises from the interpretation of coefficients involving the dependent variable, which does not affect the linearity in this case.

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RiceKernel
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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
 
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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.
 
Not helpful at all . Just had to say yes or no .
 
give a man a fish...
 
RiceKernel said:
Not helpful at all . Just had to say yes or no .
Do the work and you'll get your answer. This isn't a forum for babies, please read the rules.
 

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