# 工數 問題~及 求助~^^”

find the general solution

1.dy/dx+y=x-1/x^2

2.y'+(1/x-2)*y=3x ; y(3)=4

3.(x^2-2x)y'+(x^2-5x+4)y=(x^4-2x^3)e^-x ;y(3)=18e^-3

### 1 個解答

• 1 0 年前
最佳解答

(1)

y'+y=x-1/x^2

積分因子

I=EXP(對1積分)

I=EXP(X)

y=1/I[對I*(x-1/x^2)積分]

y=exp(-x)[對exp(x)*(x-1/x^2)積分]

y=exp(-x)[x*exp(x)-exp(x)+c+對-exp(x)*(1/x^2)積分]

y=x-1+c*exp(-x)+exp(x)*對-exp(x)*(1/x^2)積分

(-exp(x)*(1/x^2)無法積分直接寫)

(2)

y'+(1/x-2)y=3x

積分因子

I=exp(對1/x-2積分)

=exp(ln(x-2))

=x-2

y=1/I[對I*3x積分]

=1/x-2[對(x-2)*3x積分]

=1/x-2[x^3-3x^2+c]

=x^2(x-3/x-2)+c/x-2

代入y(3)=4

4=c

y=x^2(x-3/x-2)+4/x-2

(3)

(x^2-2x)y'+(x^2-5x+4)y=(x^4-2x^3)e^-x

同除(x^2-2x)

y'+(x^2-5x+4/x^2-2x)y=[x^3*(x-3)*exp(-x)/x*(x-2)]

y'+(1+(-3x+4/x*(x-2))y=x^2*exp(-x) (-3x+4/x*(x-2) =-2/x+-1/x-2)

積分因子

I=exp(對1+-(2/x)+-(1/x-2)積分)

=exp(x-2lnx-ln(x-2))

=exp(x)*(1/x^2)*(1/x-2)

y=1/I[對I*x^2*exp(-x)積分]

=(x^2*(x-2)*exp(-x))[對exp(x)*(1/x^2)*(1/x-2)*x^2*exp(-x)積分

=(x^2*(x-2)*exp(-x))[對1/x-2積分]

=(x^2*(x-2)*exp(-x))[ln(x-2)+c]

代入y(3)=18e^-3

18exp(-3)=9*exp(-3)*c

c=2

y=(x^2*(x-2)*exp(-x))[ln(x-2)+2]