 Home

Codex Wiki

List of universities

University of Oxford

Course of the University of Oxford

Undergraduate course of the University of Oxford

Mathematics course of the University of Oxford

Year 2 of the mathematics course of the University of Oxford

Differential equations 2 course of the University of Oxford

Past exam of the differential equations 2 course of the University of Oxford

2023

{c}

The <characteristic equation> of the <linear ordinary differential equation> is $r^2+2r+2=0$, with roots $-1\pm i$. A real pair of <homogeneous solutions> is therefore
$$
\boxed{y_1(x)=e^{-x}\cos x,\qquad y_2(x)=e^{-x}\sin x}.
$$
Their <Wronskian>, with the convention $W=y_1y_2'-y_1'y_2$, is
$$
W(x)=e^{-2x}\{\cos x(\cos x-\sin x)+(\sin x+\cos x)\sin x\}
=\boxed{e^{-2x}\ne0}.
$$
Thus the two functions are <linearly independent> throughout the interval and span its <homogeneous solutions>.

Solution

Past exam of the differential equations 2 course of the University of Oxford / 2023 / 1 / a / i / Solution

Past exam of the differential equations 2 course of the University of Oxford / 2023 / 1 / a / i / Solution

 Ancestors (13)

 Discussion (0)

 Articles by others on the same topic (0)

Past exam of the differential equations 2 course of the University of Oxford / 2023 / 1 / a / i / Solution

 Ancestors (13)

 Discussion (0)  Subscribe (1)

 Articles by others on the same topic (0)

 Discussion (0)