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How do you tell whether a differential equation is nonlinear or linear. And how do you determine the order and whether it is homogeneous? I'm trying to understand by reading my textbook, but I'm quite confused.

Thanks!

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The order of a differential equation is the highest derivative in equation. Th highest power of an order in equation is degree for differential equation. A differential equation is linear if we can write that of the form ‎$$f_n\frac{d^ny}{dx^n}+f_{n-1}\frac{d^{n-1}y}{dx^{n-1}}+\cdots+f_1\frac{dy}{dx}+f_0y=g$$‎ ‎\begin{eqnarray}‎ y^{\prime\prime\prime}-6xy^\prime=2-3e^x &\hspace{1cm}& \text{order(3),degree(1),linear} \\‎ y+9x(y^\prime)^2=e^x-5 &\hspace{1cm}& \text{order(1),degree(2),nonlinear} \\‎ x^2\frac{d^2y}{dx^2}-2x\frac{dy}{dx}-(\sin x)y=0 &\hspace{1cm}& \text{order(2),degree(1),linear} \\‎ y^{\prime\prime}y-2x+y^\prime=e^x &\hspace{1cm}& \text{order(2),degree(1),nonlinear} \\‎ 3t(\frac{dy}{dt})^3+4(\sin t)y^4-2=0 &\hspace{1cm}& \text{order(1),degree(3),nonlinear} \\ t^2\frac{d^3y}{dt^3}-\sin t\frac{dy}{dt}-\cos(ty)=0‎ &\hspace{1cm}& \text{order(3),degree(-),nonlinear} \\ 5x\dot{y}-4(\dot{y})^7y=x &\hspace{1cm}& \text{order(1),degree(7),nonlinear} \\ \end{eqnarray}‎

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    $\begingroup$ @projectilemotion thanks $\endgroup$ – Nosrati Jan 25 '17 at 22:12
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What is the order of the highest derivative? (i.e. first derivative is order 1, second second is order 2..) that's the order of your DE.

Look at the function in your DE (usually $y$, not sure what notation you use). Also look at the derivatives present ($y'$, $y''$, etc). Are any inside of some nonlinear function such as a power, right etc? Then the DE is nonlinear. The DE is linear if the function and all its derivatives are to the first power only.

Look at the coefficients of your function and its derivatives. If all coefficients are constant, the DE is autonomous. If any coefficient is a function of your variable, then the DE is non-autonomous.

Now look for the term which does not have your function or its derivatives in it. If it is zero, your DE is homogeneous. If it is not zero (a constant or a function of your variable) then your DE is non-homogeneous.

When beginning to save a DE, I would recommend making an abbreviated note to yourself of the type of DE. For instance, most courses spend a lot of time on SOLHA DEs: second-order linear, homogeneous, autonomous DEs. FOLHA is about as simple as they get. Those and FOLA can be solved completely, not to be said of all DEs.

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A linear differential equation is generally given by: $$\frac{d^n y}{dx^n}+f_{n-1}(x) \frac{d^{n-1} y}{dx^{n-1}}+...+f_{2}(x) \frac{d^2 y}{dx^2}+f_1(x)\frac{dy}{dx}+f_0(x)y=g(x) \tag{1}$$ Obviously, a non-linear differential equation is one which cannot be written in that form.


The order of a differential equation is the highest derivative present in the differential equation.

For example, these are examples of second order differential equations, because the $\frac{d^2y}{dx^2}$ term is the highest derivative. Note that they are also both linear, since the first one can be written in the form of $(1)$ if divided by $a$ on both sides, and the second can be divided by $x^2$: $$a \frac{d^2 y}{dx^2}+b\frac{dy}{dx}+cy=0$$ $$x^2 \frac{d^2 y}{dx^2}+x\frac{dy}{dx}+y=\cos{x}$$


A homogeneous differential equation is one which can be written in the form: $$\frac{d^n y}{dx^n}+f_{n-1}(x) \frac{d^{n-1} y}{dx^{n-1}}+...+f_{2}(x) \frac{d^2 y}{dx^2}+f_1(x)\frac{dy}{dx}+f_0(x)y=0 \tag{2}$$ Which is when $g(x)=0$.

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