6. (a) 求下列微分方程的通解：

(b) 求在 $x=0$ 时满足 $y=0$ 且 $\dfrac{\mathrm{d}y}{\mathrm{d}x}=0$ 的特解。

先解 complementary function，再设 particular integral 为 $a\cos x+b\sin x$。因为右边是 $6\cos x$，这个形式最直接。

The auxiliary equation is

So

Hence the complementary function is

For a particular integral, let

Then

Substitute into the differential equation:

Comparing coefficients,

Solving gives

Therefore the general solution is

把 $x=0$、$y=0$ 代入先求 $A$。再对通解微分，用 $x=0$、$y'=0$ 求 $B$。

Using $y=0$ at $x=0$,

so

Differentiate the general solution:

Using $\dfrac{\mathrm{d}y}{\mathrm{d}x}=0$ at $x=0$,

Since $A=-\dfrac65$,

Therefore the particular solution is

如果熟悉复指数形式，也可以把 complementary function 写成

这和上述实三角形式是同一个解空间。若熟悉 Euler formula，可以直接求 $P,Q$，最后再化回实数形式。

Write the general solution as

Using $y=0$ at $x=0$,

So

Differentiate:

Using $\dfrac{\mathrm{d}y}{\mathrm{d}x}=0$ at $x=0$,

Since $P+Q=-\dfrac65$,

Therefore,

so

Solving the two equations,

Hence,

Combining the conjugate exponential terms,

This gives the required particular solution.