I am new to the area of solving differential equations, and I came across the following differential equation and was wondering whether it was linear or non-linear:
$dy/dx= x^3 + y^3$
I would have said that the equation is non-linear, as the dependant variable being multiplied by itself three times, but the book says that the differential equation is linear?
Just out of curiosity, would the following integral be linear:
$d^4y/dx^4= x^3 + y^3$
Cheers.
On
A differential equation is called linear if it is linear with respect to the unknown function and its derivatives. For a first order equationt this means that the equation has the form: $$ y'+f(x)y+g(x)=0 $$ Your equation has the form: $$ y'-y^3-x^3=0 $$ so it is not linear.
You can see the same for the other equation.
Both of the differential equations you present are nonlinear, for the reason you cite: the presence of the $y^{3}$ term.
To add a bit more just for clarity: \begin{eqnarray} &\frac{\mathrm{d}y}{\mathrm{d}x} = x^{3}+y&\text{ is linear,}\\[6pt] &\frac{\mathrm{d}^{4}y}{\mathrm{d}x^{4}} = x^{3}+y&\text{ is linear,}\\[6pt] &\frac{\mathrm{d}y}{\mathrm{d}x} = x^{3}+y^{3}&\text{ is nonlinear,}\\[6pt] &\frac{\mathrm{d}^{4}y}{\mathrm{d}x^{4}} = x^{3}+y^{3}&\text{ is nonlinear,}\\[6pt] &\text{and also, as a bonus example,}&\\[6pt] &\left(\frac{\mathrm{d}y}{\mathrm{d}x}\right)^{4} = x^{3}+y&\text{ is nonlinear.}\\ \end{eqnarray}