高等数学考研笔记(八):微分方程

  • 可分离变量的方程:
    • 定义:

    在这里插入图片描述

    ⇒ \Rightarrow 若存在 g ( y 0 ) = 0 g(y_0) = 0 g(y0)=0,则 y = y 0 y=y_0 y=y0是方程的奇解,不包括在隐式通解中,需要单独补上;

    • 可化为分离变量方程的类型:

      • 线性组合:若微分方程形如:
        d y d x = f ( a x + b y + c ) \cfrac{dy}{dx} = f(ax+by+c) dxdy=f(ax+by+c)
        则令: u = a x + b y + c , d u = a d x + b d y u=ax+by+c,du=adx+bdy u=ax+by+c,du=adx+bdy,有:
        d u d x = a + b d y d x = a + b f ( u ) \cfrac{du}{dx} = a+b\cfrac{dy}{dx} = a+bf(u) dxdu=a+bdxdy=a+bf(u)

      • 齐次式:若微分方程形如:
        d y d x = f ( y x ) \cfrac{dy}{dx} = f(\cfrac{y}{x}) dxdy=f(xy)
        则令: u = y x , d y = u d x + x d u u=\cfrac{y}{x},dy = udx+xdu u=xy,dy=udx+xdu,有:
        d y d x = u + x d u d x = f ( u ) \cfrac{dy}{dx} = u+x\cfrac{du}{dx} = f(u) dxdy=u+xdxdu=f(u)

      • 齐次线性组合:若微分方程形如:
        d y d x = f ( a 1 x + b 1 y + c 1 a 2 x + b 2 y + c 2 ) \cfrac{dy}{dx} = f(\cfrac{a_1x+b_1y+c_1}{a_2x+b_2y+c_2}) dxdy=f(a2x+b2y+c2a1x+b1y+c1)
        则令: { u = a 1 x + b 1 y + c 1 , v = a 2 x + b 2 y + c 2 , ( a 2 , b 2 ) ≠ k ( a 1 , b 1 ) u = a 1 x + b 1 y , ( a 2 , b 2 ) = k ( a 1 , b 1 ) \begin{cases}u=a_1x+b_1y+c_1,v=a_2x+b_2y+c_2,&(a_2,b_2)\neq k(a_1,b_1)\\u=a_1x+b_1y,&(a_2,b_2)= k(a_1,b_1)\end{cases} {u=a1x+b1y+c1,v=a2x+b2y+c2,u=a1x+b1y,(a2,b2)=k(a1,b1)(a2,b2)=k(a1,b1)

  • 一阶线性微分方程:
    • 定义:形如:
      y ′ + P ( x ) y = Q ( x ) y'+P(x)y = Q(x) y+P(x)y=Q(x)
      的微分方程称为一阶线性微分方程,当 Q ( x ) = 0 Q(x)=0 Q(x)=0时称一阶齐次线性微分方程

    • 通解:令积分因子: u = e ∫ P ( x ) d x u = e^{\int P(x)dx} u=eP(x)dx,方程两边同时乘以积分因子,整理得:
      d ( y u ) = Q ( x ) × u d x d(yu) = Q(x)\times udx d(yu)=Q(x)×udx
      故通解为:
      y = 1 u ( C + ∫ Q ( x ) × u d x ) y = \cfrac{1}{u}(C+\int Q(x)\times udx) y=u1(C+Q(x)×udx)

  • 伯努利方程:
    • 定义:形如:
      d y d x + P ( x ) y = Q ( x ) y k , k ≠ 0 , 1 且 k ∈ R \cfrac{dy}{dx}+P(x)y = Q(x)y^k,k\neq 0,1且k\in R dxdy+P(x)y=Q(x)yk,k=0,1kR

    • 通解:

      y ≠ 0 y\neq 0 y=0时,两边同时乘以 y − k y^{-k} yk,得到:
      y − k d y d x + P ( x ) y 1 − k = 1 1 − k d ( y 1 − k ) d x + P ( x ) y 1 − k = Q ( x ) y^{-k}\cfrac{dy}{dx}+P(x)y^{1-k} = \cfrac{1}{1-k}\cfrac{d(y^{1-k})}{dx}+P(x)y^{1-k} = Q(x) ykdxdy+P(x)y1k=1k1dxd(y1k)+P(x)y1k=Q(x)
      u = y 1 − k u = y^{1-k} u=y1k,则上式化为一阶线性微分方程:
      u ′ + ( 1 − k ) P ( x ) u = ( 1 − k ) Q ( x ) u'+(1-k)P(x)u = (1-k)Q(x) u+(1k)P(x)u=(1k)Q(x)

  • 全微分方程:
    • 若微分方程形如:
      P ( x , y ) d x + Q ( x , y ) d y = 0 P(x,y)dx+Q(x,y)dy = 0 P(x,y)dx+Q(x,y)dy=0
      且满足: ∂ Q ∂ x = ∂ P ∂ y \cfrac{\partial Q}{\partial x} = \cfrac{\partial P}{\partial y} xQ=yP,则存在可微函数 u ( x , y ) u(x,y) u(x,y),使得 d u = P d x + Q d y du=Pdx+Qdy du=Pdx+Qdy,且有:
      u ( x , y ) = ∫ x 0 x P ( x , y 0 ) d x + ∫ y 0 y Q ( x , y ) d y = ∫ x 0 x P ( x , y ) d x + ∫ y 0 y Q ( x 0 , y ) d y u(x,y) = \int_{x_0}^xP(x,y_0)dx+\int_{y_0}^yQ(x,y)dy = \int_{x_0}^xP(x,y)dx+\int_{y_0}^yQ(x_0,y)dy u(x,y)=x0xP(x,y0)dx+y0yQ(x,y)dy=x0xP(x,y)dx+y0yQ(x0,y)dy
      其中: ( x 0 , y 0 ) (x_0,y_0) (x0,y0)可以从 P , Q P,Q P,Q的公共定义域内任取;
  • 高阶微分方程:
    • 可降阶的高阶微分方程:形如:
      y ( n ) = f ( x ) y^{(n)} = f(x) y(n)=f(x)
      ⇒ \Rightarrow 连续两边积分n次即可;

    • 不显含y的微分方程:形如:
      f ( x , y ′ , y ′ ′ ) = 0 f(x,y',y'') = 0 f(x,y,y)=0
      y ′ = p ( x ) y' = p(x) y=p(x),则 y ′ ′ = p ′ y''=p' y=p,上式化为一阶微分方程:
      f ( x , p , p ′ ) = 0 f(x,p,p') = 0 f(x,p,p)=0

    • 不显含x的微分方程:形如:
      f ( y , y ′ , y ′ ′ ) = 0 f(y,y',y'') = 0 f(y,y,y)=0
      y ′ = p ( y ) y'=p(y) y=p(y),则: y ′ ′ = d p d x = d p d y d y d x = p × p y ′ y''=\cfrac{dp}{dx} = \cfrac{dp}{dy}\cfrac{dy}{dx} = p\times p'_y y=dxdp=dydpdxdy=p×py,则上式化为一阶微分方程:
      f ( y , p , p × p ′ ) = 0 f(y,p,p\times p') = 0 f(y,p,p×p)=0

  • 二阶线性微分方程:
    • 定义:形如:
      y ′ ′ + p ( x ) y ′ + q ( x ) y = f ( x ) y''+p(x)y'+q(x)y = f(x) y+p(x)y+q(x)y=f(x)
      的方程叫二阶线性微分方程,当 f ( x ) = 0 f(x)=0 f(x)=0时,称二阶齐次线性微分方程

    • 齐次方程的通解结构:若 y 1 ( x ) , y 2 ( x ) y_1(x),y_2(x) y1(x),y2(x)是二阶方线性微分方程的两个线性无关的解,则:
      y = C 1 y 1 + C 2 y 2 y = C_1y_1+C_2y_2 y=C1y1+C2y2
      即是该方程的通解,其中 C 1 , C 2 C_1,C_2 C1,C2是互相独立的任意常数;

    • 非齐次方程的通解结构:

      y ∗ y^* y是二阶非齐次线性微分方程的特解,而 C 1 y 1 + C 2 y 2 C_1y_1+C_2y_2 C1y1+C2y2是对应齐次线性微分方程的通解,则:
      y = C 1 y 1 + C 2 y 2 + y ∗ y = C_1y_1+C_2y_2 + y^* y=C1y1+C2y2+y
      即是非齐次方程的通解;

      ⇒ \Rightarrow y 1 ∗ y_1^* y1 y 2 ∗ y_2^* y2是非齐次方程的两个特解,则 y 1 ∗ − y 2 ∗ y_1^*-y_2^* y1y2是对应齐次方程的通解;

      ⇒ \Rightarrow y 1 y_1 y1 y 2 y_2 y2分别是非齐次方程 y ′ ′ + p ( x ) y ′ + q ( x ) y = f 1 ( x ) y''+p(x)y'+q(x)y = f_1(x) y+p(x)y+q(x)y=f1(x) y ′ ′ + p ( x ) y ′ + q ( x ) y = f 2 ( x ) y''+p(x)y'+q(x)y = f_2(x) y+p(x)y+q(x)y=f2(x)的两个解,则 y 1 + y 2 y_1+y_2 y1+y2是方程 y ′ ′ + p ( x ) y ′ + q ( x ) y = f 1 ( x ) + f 2 ( x ) y''+p(x)y'+q(x)y = f_1(x)+f_2(x) y+p(x)y+q(x)y=f1(x)+f2(x)的通解;

    • 常数变易法:若 y 1 , y 2 y_1,y_2 y1,y2是齐次方程的两个线性无关的解,则其非齐次方程有特解:
      y ∗ = C 1 ( x ) y 1 + C 2 ( x ) y 2 y^* = C_1(x)y_1+C_2(x)y_2 y=C1(x)y1+C2(x)y2
      其中, C 1 ( x ) , C 2 ( x ) C_1(x),C_2(x) C1(x),C2(x)满足方程组:
      { C 1 ′ y 1 + C 2 ′ y 2 = 0 C 1 ′ y 1 ′ + C 2 ′ y 2 ′ = f ( x ) \begin{cases} C_1'y_1 + C_2'y_2 = 0\\ C_1'y_1'+C_2'y_2' = f(x) \end{cases} {C1y1+C2y2=0C1y1+C2y2=f(x)

  • n阶线性常系数微分方程:
    • 定义:形如:
      y ( n ) + c 1 y ( n − 1 ) + . . . + c n y = f ( x ) y^{(n)}+c_1y^{(n-1)}+...+c_ny = f(x) y(n)+c1y(n1)+...+cny=f(x)
      的方程称为n阶线性常系数微分方程,当 f ( x ) = 0 f(x)=0 f(x)=0时称n阶齐次线性常系数微分方程

    • 特征方程:
      λ n + c 1 λ n − 1 + . . + c n = 0 \lambda^n+c_1\lambda^{n-1}+..+c_n = 0 λn+c1λn1+..+cn=0

    • 齐次方程通解的结构:

    在这里插入图片描述

    • 待定系数法求非齐次方程特解:

      • f ( x ) = P m ( x ) e α x f(x) = P_m(x)e^{\alpha x} f(x)=Pm(x)eαx,其中 P m ( x ) P_m(x) Pm(x)是关于x的m次多项式函数,则非齐次方程的特解形式为:
        y ∗ = x k Q m ( x ) e α x y^* = x^kQ_m(x)e^{\alpha x} y=xkQm(x)eαx
        其中,k是 α \alpha α作为特征方程 λ 2 + p λ + q = 0 \lambda^2+p\lambda+q=0 λ2+pλ+q=0的特征根的重数, Q m ( x ) Q_m(x) Qm(x)是关于x的m次多项式函数;

      • f ( x ) = [ A m 1 ( x ) c o s β x + B m 2 ( x ) sin ⁡ β x ] e α x f(x) = [A_{m_1}(x)cos\beta x+B_{m_2}(x)\sin\beta x]e^{\alpha x} f(x)=[Am1(x)cosβx+Bm2(x)sinβx]eαx,其中 A m 1 ( x ) A_{m_1}(x) Am1(x) B m 2 ( x ) B_{m_2}(x) Bm2(x)分别是x的 m 1 m_1 m1 m 2 m_2 m2次多项式,令 m = m a x { m 1 , m 2 } m=max\{m_1,m_2\} m=max{m1,m2},则非齐次方程的特解形式为:
        y ∗ = x k [ C m ( x ) c o s β ( x ) + D m s i n β x ] e α x y^* = x^k[C_m(x)cos\beta (x)+D_m sin \beta x]e^{\alpha x} y=xk[Cm(x)cosβ(x)+Dmsinβx]eαx
        其中,k是 α + β i \alpha+\beta i α+βi作为特征方程 λ 2 + p λ + q = 0 \lambda^2+p\lambda+q=0 λ2+pλ+q=0的特征根的重数, C m ( x ) , D m ( x ) C_m(x),D_m(x) Cm(x),Dm(x)均是关于x的m次多项式函数;

  • 二阶欧拉方程:
    • 定义:形如:
      x 2 y ′ ′ + p x y ′ + q y = f ( x ) x^2y''+pxy'+qy = f(x) x2y+pxy+qy=f(x)

    • 通解:令 x = e t x = e^t x=et D = d d t D = \cfrac{d}{dt} D=dtd则:
      d y d x = d y d t d t d x = 1 x D y d 2 y d x 2 = − 1 x 2 D y + 1 x 2 D 2 y = 1 x 2 [ D ( D − 1 ) ] y \cfrac{dy}{dx} = \cfrac{dy}{dt}\cfrac{dt}{dx}=\cfrac{1}{x}Dy\\ \cfrac{d^2y}{dx^2}=-\cfrac{1}{x^2}Dy+\cfrac{1}{x^2}D^2y = \cfrac{1}{x^2}[D(D-1)]y dxdy=dtdydxdt=x1Dydx2d2y=x21Dy+x21D2y=x21[D(D1)]y
      故原方程化为二阶常系数线性微分方程:
      [ D ( D − 1 ) ] y + p D y + q y = f ( e t ) [D(D-1)]y+pDy+qy = f(e^t) [D(D1)]y+pDy+qy=f(et)

    • 推广:对于n阶欧拉方程
      x n y ( n ) + c 1 x n − 1 y ( n − 1 ) + . . + c n − 1 x y ′ + c n y = f ( x ) x^ny^{(n)}+c_1x^{n-1}y^{(n-1)}+..+c_{n-1}xy'+c_ny = f(x) xny(n)+c1xn1y(n1)+..+cn1xy+cny=f(x)
      同理令 x = e t x=e^t x=et D = d d t D = \cfrac{d}{dt} D=dtd,可得: x k y ( k ) = D ( D − 1 ) . . ( D − k + 1 ) y = D ! ( D − k ) ! y = A D k y x^{k}y^{(k)} = D(D-1)..(D-k+1)y = \cfrac{D!}{(D-k)!}y = A_D^k y xky(k)=D(D1)..(Dk+1)y=(Dk)!D!y=ADky

      原方程化为n阶常系数微分方程:
      ∑ k = 0 n c k A D k y = f ( x ) ,   c 0 = 1 \sum\limits_{k=0}^n c_k A_D^k y = f(x),\space c_0 = 1 k=0nckADky=f(x), c0=1



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