Question

求微分方程(x-2)dy/dx=y+2(x-3)^3的通解，在线等

Answer 1

求微分方程(x-2)dy/dx=y+2(x-3)³的通解
解：先求 (x-2)dy/dx=y的通解：分离变量得 dy/y=dx/(x-2);
积分之得lny=ln(x-2)+lnc₁=ln[c₁(x-2)]
∴y=c₁(x-2)；将c₁换成x的函数u，得y=u(x-2)............①
对①取导数得：dy/dx=u+(x-2)du/dx...........②
将①②代入原式得：(x-2)[u+(x-2)du/dx]=u(x-2)+2(x-3)³
消去同类项得：(x-2)²du/dx=2(x-3)³；分离变量得du=[2(x-3)³/(x-2)²]dx
积分之得: u=2∫[(x-3)³/(x-2)²]dx=2∫[(x-2)-1]³dx/(x-2)²
=2∫[(x-2)³-3(x-2)²+3(x-2)-1]dx/(x-2)²=2∫[(x-2)-3+3/(x-2)-1/(x-2)²]dx
=2[(1/2)(x-2)²-3x+3ln(x-2)+1/(x-2)]+c=(x-2)²-6x+6ln(x-2)+2/(x-2)+c
=x²-10x+6ln(x-2)+2/(x-2)+C【其中C=4+c】........③
将③代入①式即得原方程的通解为：y=(x-2)[x²-10x+6ln(x-2)+2/(x-2)+C]

Answer 2

解：∵(x-2)dy/dx=y+2(x-3)^3
==>(x-2)dy-ydx=2(x-3)^3dx
==>((x-2)dy-ydx)/(x-2)^2=2(x-3)^3dx/(x-2)^2 (等式两端同除(x-2)^2)
==>d(y/(x-2))=(2x-10+6/(x-2)-2/(x-2)^2)dx
==>∫d(y/(x-2))=∫(2x-10+6/(x-2)-2/(x-2)^2)dx
==>y/(x-2)=x^2-10x+6ln│x-2│+2/(x-2)+C (C是积分常数)
==>y=(x-2)(x^2-10x)+6(x-2)ln│x-2│+2+C(x-2)
==>y=x^3-12x^2+20x+2+6(x-2)ln│x-2│+C(x-2)
∴原方程的通解是y=x^3-12x^2+20x+2+6(x-2)ln│x-2│+C(x-2)。

Answer 3

(x-2)dy/dx=y+2(x-3)^3
（x-2）y'-y=2(x-3)^3
非齐次非常系数微分方程。对应齐次微分方程：
（x-2）y'-y=0
y'/y=1/(x-2)
(lny)'=(ln(x-2))'
lny=ln(x-2)+C1=ln[C2(x-2)]
y=C2(x-2)
假设常数C2也是x的函数，上述函数是原来非齐次方程的特解：
y'=C2'(x-2)+C2
(x-2)[C2'(x-2)+C2]-C2(x-2)=2(x-3)^3
C2'(x-2)²=2(x-3)^3
C2'=2(x-3)^3/(x-2)²
=2[x³-9x²+27x-27]/(x²-4x+4)
做长除法：
（1，-4,4）|（1，-9,27，-27）
-----------1|（1，-4,4）
-------------------（-5,23，-27）
............................-5| （-5,20，-20）
------------------------（3，-7）
C2'/2=x-5+(3x-7)/(x-2)²
=x-5+(3x-6-1)/(x-2)²
=x-5+3/(x-2)-1/(x-2)²
积分
C2/2=x²/2-5x+3ln(x-2)+1/(x-2)+C3
C2=x²-10x+6ln(x-2)+2/(x-2)+2C3
y=C2(x-2)+(x-2)(x²-10x+6ln(x-2)+2/(x-2)+2C3)
=(x-2)[x²-10x+6ln(x-2)+2/(x-2)+C2+2C3]
设C=C2+2C3

=(x-2)[x²-10x+6ln(x-2)+2/(x-2)+C]
=C(x-2)+x²(x-2)-10x(x-2)+6(x-2)ln(x-2)+2
=Cx-2C+x³-2x²-10x²+20x+2+6(x-2)ln(x-2)
=x³-12x²+（20+C）x+2（1-C）+6(x-2)ln(x-2)
验证：

y'=3x²-24x+（20+C）+6ln(x-2)+6
=3x²-24x+（26+C）+6ln(x-2)
左边=（x-2）y'
=（x-2）[3x²-24x+（26+C）+6ln(x-2)]
=3x³-24x²+（26+C）x+6（x-2）ln(x-2)-6x²+48x-2（26+C）
=3x³-30x²+（74+C）x-2（26+C）+6（x-2）ln(x-2)
右边=y+2(x-3)^3
=x³-12x²+（20+C）x+2（1-C）+6(x-2)ln(x-2)+2（x³-9x²+27x-27）
=3x³-30x²+（74+C）x+2（-26-C）+6(x-2)ln(x-2)
=3x³-30x²+（74+C）x-2（26+C）+6(x-2)ln(x-2)
正确

Answer 4

解：∵(x-2)dy/dx=y+2(x-2)^3 ==>(x-2)dy-ydx=2(x-2)^3dx ==>dy/(x-2)-ydx/(x-2)^2=2(x-2)dx (等式两端同除(x-2)^2) ==>d(y/(x-2))=d((x-2)^2) ==>y/(x-2)=(x-2)^2+C (C是常数) ==>y=(x-2)^3+C(x-2) ∴原方程的通解是y=(x-2)^3+C(x-2)。

Answer 5

当X-2不等于0时，变换为：dy/dx-y/（X-2)=2(X-3)^3/(X-2),这是标准型一阶微分方程，P（x）=-1/(X-2),Q（x）=2(X-3)^3/(X-2),先求齐次通解。y1=C(X-2),后面的把2(X-3)^3/(X-2)化简的到2（x-2）^2-2/(x-2)-6(x-2)+6就可以积分了，得到y*=2（x-2)^3/3-2ln!x-2!-3(x-2)^2+6x,通解为y=y1+y*=C(X-2)+2（x-2)^3/3-2ln!x-2!-3(x-2)^2+6x，你自己化简得了！！x-2!代表x-2的绝对值