SOLVE THE DIFFERENTIAL EQUATION AND CAUCHY PROBLEMS BELOW:
(Đã check bằngMathematica 9)
A. REDUCED DIFFERENTIAL EQUATION LEVEL 2
1) y’’ –
𝑦
𝑥−1 - x(x - 1) = 0; y(2) = 1; y’(2) = -1
Sol 𝒚
𝟏
𝟐𝟒
𝟑
𝒙
𝟒−𝟒
𝒙
𝟑−𝟑𝟔
𝒙
𝟐+𝟕𝟐𝒙+𝟖
𝑪
𝟏=−𝟑
𝑪
𝟐=𝟏/𝟑
2) y’’ + 2y’(1 – 2y) = 0; y(0) = 0; y’ (0) =
1
2

Sol 𝒚
𝒙
𝟐(𝒙+𝟏
𝑪
𝟏=𝟏
𝑪
𝟐=𝟏/𝟐
3) xy’’ – y’ = x2.lnx; y(1) =
4
9; y’(1) = -1
Sol 𝒚
𝒙
𝟑
𝟑
𝐥𝐧
𝒙−𝟒
𝟗
𝑪
𝟏=𝟎
𝑪
𝟐=𝟎
B. HOMOGENOUS EQUATION
1) x(x – 1)2.y’’ + x(x - 1).y’ – y = 0;
known that y = ( +
𝛽
1−𝑥 is a separate test.
Sol 𝒚
𝑪
𝟏
𝒙
𝒙−𝟏
𝑪
𝟐
𝒙
𝐥𝐧
𝒙+𝟏
𝒙−𝟏

2) y’’ + y’.tgx – y.cos2x = 0;
known that y = e(sinx is a separate test.
Sol 𝒚
𝑪
𝟏
𝒆
𝐬𝐢𝐧
𝒙
𝑪
𝟐−𝟏
𝟐
𝒆
𝐬𝐢𝐧
𝒙

3) (x2 - 1).y’’ + 2xy’ = 0.
Sol 𝒚
𝑪
𝟏
𝑪
𝟐
𝐥𝐧
𝟏−𝒙
𝟏+𝒙

4) x2(lnx - 1).y’’ – xy’ + y = 0;
known that y = x( is a separate test.
Sol 𝒚
𝑪
𝟏
𝒙
𝑪
𝟐
𝐥𝐧
𝒙

5) (2x + 1).y’’ + (4x - 2).y’ – 8y = 0;
known that y = e(x is a separate test.
Sol 𝒚
𝑪
𝟏
𝒆−𝟐𝒙
𝑪
𝟐
𝟒
𝒙
𝟐+𝟏

C. GENERAL EQUATION
1) (1 + x2).y’’ + 2xy’ – 2y = 4x2 + 2;
known that y = x2 is a separate test.
Sol 𝒚
𝑪
𝟏
𝒙
𝑪
𝟐
𝟏+𝒙
𝐭𝐚𝐧−𝟏
𝒙
𝒙
𝟐

2) (2x – x2).y’’ + 2(x - 1).y’ – 2y = -2;
known that y = x and y = 1 are separate tests.
Sol 𝒚
𝑪
𝟏
𝒙−𝟏
𝑪
𝟐
𝒙
𝟐−𝒙+𝟏+𝒙
3) x(x + 1).y’’ + (x + 2).y’ – y = x +
1
𝑥;
known that y = x + 2 is a separate test of homogenous eq.
Sol 𝒚
𝐥𝐧
𝒙
𝟏
𝒙+𝟏
𝒌
𝟏
𝒙+𝟐−𝟐
𝒙+𝟏
𝒙
𝟐
𝟐
𝒌
𝟐−𝟏
𝟐𝒙

4) x(x2 + 3).y’’ – 2(x2 + 3).y’ + 6xy = e-x(x4 + x3 + x2 – 9x – 12);
known that y = (ax + b).e-x is a separate test and y = x3 is a separate test of homogenous eq.
Saiđề???
D. LINEAR DIFFERENTIAL EQUATIONS LEVEL 2 WITH CONSTANT COEFFICIENTS 1) y’’ – y =
𝑒
𝑥
𝑒
𝑥+1

2) y’’+ y = tgx
3) y’’ + 5y’ + 6y =
1
1
𝑒
2𝑥

4) y’’ – 7y’ + 6y = sinx
5) y’’ + y = x.ex + 2.e-x
6) y’’ – y’ = e2x + ex + x