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, .

Simplify the given equation to get,

\(\begin{aligned}{c}u + (v + uv - u{e^u})\frac{{du}}{{dv}} &= 0\\(v + uv - u{e^u})\frac{{du}}{{dv}} + u &= 0\end{aligned}\)

But the term \({a_1}\) has dependence on both \(u\) and \(v\). So, the given is nonlinear in \(u\).

As, by the classification of linearity, the given differential equation should be in the form, \({a_1}(u)\frac{{dv}}{{du}} + {a_0}(u)v = g(u)\).

Simplify the given equation to get,

\(\begin{aligned}{c}u\frac{{dv}}{{du}} + (v + uv - u{e^u}) &= 0\\u\frac{{dv}}{{du}} + (1 + u)v &= u{e^u}\end{aligned}\)

It is linear in \(v\) with the parameters equal to,

\(\begin{aligned}{c}{a_1} &= u\\{a_0} &= 1 + u\end{aligned}\)

And \(g(u) = u{e^u}\).