Page 27 - Pra U STPM 2022 Penggal 2 - Mathematics

Mathematics Semester 2 STPM Chapter 4 Differential Equations

1
–∫
dx
Multiplying both sides of the differential equation with the integrating factor is e x .
dy e – —dx 2 – 1 e – —dx 2 y = –2x e – —dx 2
1
1
1
1
1
∫ x
∫ x
∫ x
dx x 1
d ye – —dx 2 = –2xe – —dx
1
1
1
∫ x
∫ x
dx
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1
1
1
∫ x
∫ x
= –2xe
\ ye – —dx 2 ∫ – —dx dx
∫
–1
–1
\ y(x ) = – 2x(x ) dx
∫
yx = –2 dx
–1
–1
yx = –2x + A
\ y = –2x + Ax
2
Example 9
dy
Find the general solution to the differential equation = x – 2y.
dx
dy
Solution: = x – 2y
dx
dy
Arrange the differential equation to the form dx + f(x)y = g(x).
dy
+ 2y = x
dx
\ f(x) = 2 and g(x) = x.
4
So the integrating factor is e ∫ 2 dx = e .
2x
Multiplying both sides of the equation with e ;
2x
dy (e ) + 2(e )y = x(e )
2x
2x
2x
dx
\ d (ye ) = xe 2x
2x
dx
∫
2x
\ ye = xe dx + C
2x
Integrating the right side by parts:
du = e and v = x
2x
dx
∫
2x
\ u = e dx = 1 e and dv = 1
2x
2
dx
1
∫
\ ye = 1 xe – e dx + C
2x
2x
2x
2 2
\ ye = 1 xe – 1 e + C
2x
2x
2x
2 4
\ y = 1 x – 1 + Ce –2x
2 4
135
04 STPM Math(T) T2.indd 135 28/01/2022 5:44 PM