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y ′ + y = sin x<br />
y IV − 16y = 1 + cos 2x<br />
y ′′′ − 6y ′′ + 11y ′ − 6y = x 2 <br />
y IV − y ′′ = x − 1<br />
<br />
y ′ + y = 0 y0(x) = ce −x c ∈ R <br />
sin x <br />
A sin x + B cos x <br />
A cos x − B sin x + A sin x + B cos x = sin x A + B = 0 A − B = 1<br />
A = −B = 1/2 y(x) = ce −x + (sin x − cos x)/2 c ∈ R<br />
y IV −16y = 0 λ 4 −16 = 0 <br />
{±2, ±2i} y0(x) = c1e 2x +c2e −2x +c3 cos(2x)+<br />
c4 sin(2x) ci ∈ R i = 1, 2, 3, 4 1 + cos 2x <br />
¯y1(x) y IV − 16y = 1 ¯y2(x) <br />
y IV − 16y = cos 2x y IV − 16y = 1 <br />
0 <br />
0 ¯y1(x) =<br />
−1/16<br />
y IV −16y = cos 2x <br />
2 1 <br />
¯y(x) = x(A cos(2x) + B sin(2x)) <br />
<br />
¯y ′ (x) = A cos(2x) + B sin(2x) + 2x(−A sin(2x) + B cos(2x))<br />
¯y ′′ (x) = 2(−A sin(2x) + B cos(2x)) + 2(−A sin(2x) + B cos(2x)) + 4x(−A cos(2x) − B sin(2x))<br />
= 4(−A sin(2x) + B cos(2x)) + 4x(−A cos(2x) − B sin(2x))<br />
¯y ′′′ (x) = 8(−A cos(2x) − B sin(2x)) + 4(−A cos(2x) − B sin(2x)) + 4x(A sin(2x) − B cos(2x))<br />
= 12(−A cos(2x) − B sin(2x)) + 8x(A sin(2x) − B cos(2x))<br />
¯y IV (x) = 24(A sin(2x) − B cos(2x)) + 8(A sin(2x) − B cos(2x)) + 16x(A cos(2x) + B sin(2x))<br />
= 32(A sin(2x) − B cos(2x)) + 16x(A cos(2x) + B sin(2x))<br />
<br />
32(A sin(2x) − B cos(2x)) + 16x(A cos(2x) + B sin(2x)) − 16x(A cos(2x) + B sin(2x)) = cos 2x<br />
A = 0 −32B cos(2x) = cos 2x B = −1/32 <br />
<br />
y(x) = c1e 2x + c2e −2x + c3 cos(2x) + c4 sin(2x) − 1 x<br />
−<br />
16 32 sin(2x),<br />
ci ∈ R i = 1, 2, 3, 4
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