I have a problem understanding how to define a linear or non-linear Differential equation. These are my answers to the questions, however, my teacher's answers are not the same as mine.
Questions
His answers are
1.Linear
2.Non-Linear
3.Non-Linear
4.Linear
5.Non-Linear
6.Linear
Isn't the right-hand side of the equation has to be function of x or y alone? I extremely confused now, please help.
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$\begingroup$In an ODE $F(x,y,y',y'',..)=0$, if $y, y', y'', ...$ occur linearly (power 1) it can be linear. For example $y'=x+y^2, y'=2x+e^y, y'=3x+\sin(y),...$ are not linear. $y'^2=x+y$ is not linear.
On the other hand$a(x)y''+b(x)y'+c(x)y=0$ is linear. It being second order if $y_1(x)$ and $y_2(x)$are its solution then $y(x)=y_1(x)+y(x)$ will always be a solution.
Suppose $y_1(x)$ and $y_2(x)$ are solutions of an ODE then if $y(x)=C_1 y_1+C_2 y_2$ is also a solution, the ODE is a linear ODE. This can serve as a general method to check if a given ODE is linear or not.
Check that, despite $y$ and $y'$ occuring linearly the ODE: $(x+y)dx+(y-x)dy=0$or equivalenly $y'=\frac{x-y}{x+y}$ is not linear.
$\endgroup$ $\begingroup$Hint:
A linear differential equation has the form
$$ c_0(x)y + c_1(x)\frac{d{y}}{dx} + \cdots c_k(x) \frac{d^ky}{dx^k} + \alpha(x) = 0 $$
where the $c_i(x)$ and $\alpha(x)$ are differentiable.
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