Solution of Exam on Ordinary Differential Equations. Trial Aleph ...
Solution of Exam on Ordinary Differential Equations. Trial Aleph ...
Solution of Exam on Ordinary Differential Equations. Trial Aleph ...
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4.1.1 General soluti<strong>on</strong>. The …rst way<br />
y 0 + y = e x y 3 , y0<br />
y<br />
+ 1 3 y<br />
= e x<br />
2<br />
z = 1<br />
y<br />
) z 0 = 2y0<br />
2 y<br />
, y0<br />
3 y<br />
= 1 3 2 z0<br />
1<br />
2 z0 + z = e x , z 0 2z = 2e x<br />
z 0 2z = 2e x (26)<br />
Ce<br />
z = e 2x C 2 R e 2x e x dx = e 2x C 2 R e x dx = Ce 2x + 2e 2x e x =<br />
2x + 2e x z = Ce 2x + 2e x (27)<br />
z = 1<br />
y 2 , 1<br />
y 2 = Ce 2x + 2e x , y 2 =<br />
1<br />
Ce 2x +2e x<br />
y 2 =<br />
1<br />
Ce 2x + 2e x ; C 2 R (28)<br />
lim y (x) = lim p 1<br />
C!1 C!1 Ce<br />
= 0;so y (x) = 0 is a partial soluti<strong>on</strong>.<br />
2x +2e x<br />
1<br />
y (x) = p<br />
Ce<br />
2x<br />
+ 2e ; C 2 R<br />
x<br />
4.1.2 General soluti<strong>on</strong>. The sec<strong>on</strong>d way<br />
y (x) = u (x) v (x) ; y 0 = u 0 v + uv 0 ; u 0 v + uv 0 ; y 0 + y = u 0 v + uv 0 + uv = e x u 3 v 3<br />
u 0 v + u (v 0 + v) = e x u 3 v 3<br />
v 0 + v = 0 ) v = e x<br />
u 0 v = e x u 3 v 3 ; y 6= 0 ) v 6= 0 , u 0 = e x u 3 v 2 = e x u 3 e 2x = u 3 e x<br />
du<br />
dx = u3 e x ; du<br />
u<br />
1<br />
3<br />
C;<br />
e x +C = 2u2 ;<br />
u 0 = u 3 e x<br />
= e x dx; R du<br />
u 3 = R e x dx;<br />
1<br />
2u 2 = e x + C;<br />
1<br />
2u 2 = e x +<br />
u 2 =<br />
1<br />
2(e x +C) ; y2 = u 2 v 2 = u 2 e 2x =<br />
1<br />
2e 2x (e x +C) = 1<br />
2e x +Ce 2x ;<br />
y =<br />
1<br />
p<br />
2ex + Ce 2x<br />
4.1.3 Partial soluti<strong>on</strong> satisfying given initial c<strong>on</strong>diti<strong>on</strong>s<br />
y (0) = 1 2 : 1 4 = 1<br />
C+2 ; C + 2 = 4; C = 2<br />
y (x) =<br />
1<br />
p<br />
2 (e<br />
2x<br />
+ e x )<br />
5