What is a 'Linear' Differential Equation?

This video explores what it means for a differential equation to be linear. Specifically we discuss the importance of linear superposition and give examples of linear and nonlinear operators and differential equations.

@eigensteve on Twitter

This video was produced at the University of Washington

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0:00 Example of linear superposition of solutions to an ODE
6:53 Linear systems of differential equations
8:45 Examples of linear operators

[willykitheka]

Arguably, one of the best teachers alive and on the internet today! This man teaches very uniquely!

[tommyhuffman]

These videos are so valuable. Differential Equations really have the power to shape our world!

[richardcasey]

We also see linear operators in tensor calculus and tensor analysis on manifolds. His presentations are among the best at explaining these concepts.

[jamesjohn]

Thanks a lot, for great clarity of what it means for equation to be linear. I Like maths and sometimes, confused and get my mind spinning of what the object naming language means about something mathematically.

[himanshuraj]

thanks sir...these videos is helping me a lot in my Undergraduate courses now

[tascker]

These lectures are so interesting and helpful, thanks a lot.

[schenzur]

This is an amazing lecture. Thank you so much professor!

[ShySteven]

Very clear. By the way, can I ask what is the tool you use for the mirror blackboard?

[johnstuder]

Thank you for the dedicated video for linearity…but still confused. I understand that addition and multiplication are the rules that make Linear Algebra possible, and that the linearity test uses these rules…but the confusing part is the superposition of highly non-linear functions (sin/cos, e^x), right after saying they had to be linear! Clearly I am missing something fundamental. I have been hung up on this for years. Same thing applies to Matricies. How can polynomials, sun/cos and e^x be basis, when they are not linear? What am I missing? Also, you seemed to use the linearity of the function and linearity of the operator interchangeably…f(x) vs d/dt…

[leolima]

I wonder if fractionary diferential equations, even if they do not involve non-linear composition of the variable and it's fractionary and/or it's integer derivatives, are to be considered linear ...

[mariarahelvarnhagen]

Explain Thoroughly How First Order Axis X Limits A 3.14 Value For Pi But Third Order Axis Z Limits A Value Of 2 ?

[richardwright]

So, only homogeneous ode are linear? As soon as there's a constant in these or otherwise linear functions f(x), then superposition doesn't work. 0+0=0, but not 1+1=1
So linear means proportional. Just because a function is a straight line doesn't mean it's linear.
But co-ordinates can be transformed so f(0)=0
Some non-linear functions or ode s can be made linear?

[naimaware]

what about ODE with non constant parameters?