Question

求微分方程

求微分方程

Answer 1

解:微分方程为x²d²y/dx²-xdy/dx-3y=0，化为x²y"-xy'-3y=0，x²y"+2xy'-3xy'-3y=0，x²y"+2xy'-3(xy'+y)=0，(x²y')'-3(xy)'=0，x²y'-3xy=a，y'/x³-3y/x⁴=a/x⁵，设a=-4b，微分方程化为(y/x³)'=-4b/x⁵，y/x³=b/x⁴+c，微分方程为y=b/x+cx³(b、c为任意常数)方法2微分方程为x²y"-xy'-3y=0，设微分方程的特解为y=xⁿ，将特解代入微分方程中，有x²×n(n-1)xⁿ⁻²-x×nxⁿ⁻¹-3xⁿ=0，n(n-1)-n-3=0，n²-2n-3=0，得:n=3或-1，则微分方程的特解有y=x³或1/x设微分方程的通解为y=u/x，有x²(u/x)"-x(u/x)'-3y=0，x²(u"/x-2u'/x²+2u/x³)-x(u'/x-u/x²)-3u/x=0，xu"-2u'+2u/x-u'+u/x-3u/x=0，xu"-3u'=0，u"/x³-3u'/x⁴=0，(u'/x³)'=0，u'/x³=4a (a为任意常数)，u'=4ax³，u=ax⁴+c (c为任意常数)，微分方程的通解为y=ax³+c/x

Answer 2

x^2.y''-xy'-3y=0

x^2.y''+2xy' - 3xy'-3y=0

x^2.y''+2xy' = 3xy' +3y

d/dx ( x^2.y')  = d/dx (3xy)

x^2.y'   = 3xy +C1'

x^2.y' - 3xy  =C1'

y' - (3/x)y  =C1'/x^2

p(x) = -3/x

∫ p(x) dx = ∫ -(3/x) dx = -3lnx+C

e^[∫ p(x) dx] = 1/x^3

两边乘以 1/x^3

(1/x^3)[y' - (3/x)y]  =(C1'/x^2)(1/x^3)

d/dx (y/x^3) = C1'/x^5

y/x^3 =∫  C1'/x^5 dx

           = -(C1'/4)(1/x^4) + C2

y=-(C1'/4)(1/x) + C2.x^3

   =C1/x + C2.x^3

Answer 3

此乃欧拉方程。令 x = e^t, 则
dy/dx = (dy/dt)/(dx/dt) = e^(-t)dy/dt
d^2y/dx^2 = {d[e^(-t)dy/dt]/dt}/(dx/dt)
= e^(-t)[-e^(-t)dy/dt+e^(-t)d^2y/dt^2]
= e^(-2t)[-dy/dt+d^2y/dt^2]
代入微分方程得-dy/dt+d^2y/dt^2 - dy/dt - 3y = 0
d^2y/dt^2 - 2dy/dt - 3y = 0
特征方程 r^2-2r-3 = 0, r = -1, 3
y = C1e^(-t) + C2e^(3t)
= C1/x + C2x^3

Answer 4

我做一题。
由(x^2+1)y'+2xy=0得
dy/y=-2xdx/(x^2+1),
积分得lny=-ln(x^2+1)+lnc
∴y=c/(x^2+1).
设y=c(x)/(x^2+1)是(x^2+1)y'+2xy=4x^2①的解，则
y'=[(x^2+1)c'(x)-2xc(x)]/(x^2+1)^2,
代入①得c'(x)=4x^2,
∴c(x)=4x^3/3+c,
∴①的解是y=(4x^3/3+c)/(x^2+1).