91影视

If\(F\)is linear in \(y,y',...,{y^n}\), then the \({n^{th}}\)order ordinary differential equation is said to be linear. The form of the equation is given by,

\({a_n}(x)\frac{{{d^n}y}}{{d{x^n}}} + {a_{n - 1}}(x)\frac{{{d^{n - 1}}y}}{{d{x^{n - 1}}}} + L + {a_1}(x)\frac{{dy}}{{dx}} + {a_0}(x)y = g(x)\)

As, by the classification of linearity, the given differential equation should be in the form, \({a_2}(r)\frac{{{d^2}u}}{{d{r^2}}} + {a_1}(r)\frac{{du}}{{dr}} + {a_0}(r)u = g(r)\), but the term \((cos(r + u))\) has dependence both on \(u\) and \(r\). So, the given is nonlinear.

If the given equation is linear, then the term \((cos(r + u))\) must be dependent on \(r\). Because of \(\frac{{{d^2}u}}{{d{r^2}}}\), the equation is second order differential equation.