数学物理方程2new

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1、1nÙÅħ(Ü)¯1Å{§3.1ÄS3.1Åħܯ1"S31.(e¯)µutt−9uxx=0,−∞0,(1)2u(x,0)=sin2x,ut(x,0)=6x−1,−∞0,(2)2u(x,0)=sinx,ut(x,0)=x,−∞0,(3)2u(x,0)=x,ut(x,0)=1,−∞0,(4)u(x,0)=x,ut(x,0)=sinx,−∞

2、¤"a=3,φ(x)=sinx,ψ(x)=6x−1.dD’Alembertúª,Zx+3t11u(x,t)=[φ(x+3t)+φ(x−3t)+ψ(ξ)dξ26x−3tZx+3t112=(sin2(x+3t)+sin2(x−3t))+(6ξ−1)dξ26x−3t23=sin2xcos3t+6(xt+3t)−t2(2).φ(x)=sinx,ψ(x)=x.dD’Alembertúª,2123u(x,t)=sinxcosat+xt+at.32(3).a=1,φ(x)=x,ψ(x)=1,f(x,t)=1.dKirchhoffúª,ZZZx+ttx+(t+τ)111u(x,t)

3、=[φ(x+t)+φ(x−t)]+ψ(ξ)dξ+f(ξ,τ)dξdτ22x−t20x−(t−τ)232=x+t+t2(4).a=1,φ(x)=x,ψ(x)=sinx,f(x,t)=cos(x+t).dKirchhoffúª,ZZZx+ttx+(t+τ)111u(x,t)=[(x+t)+(x−t)]+sinξdξ+cos(ξ+τ)dξdτ22x−t20x−(t−τ)11=x+sinxsint+tsin(x+t)22213ÙÅħ(Ü)¯1Å{2.kÑ.«þÅħ½)¯utt−uxx=0,x>0,t>0,u(x,0)=ut(x,0)=0,x>0,tu(0,

4、t)=,t>01+t),,y²é?¿c>0,4x→lim+∞u(cx,x)3,¿ÑT4.t)µu(x,t)=v(x,t)+1+t.v(x,t)÷vvtt−vxx=2,x>0,t>0,(1+t)3v(x,0)=0,vt(x,0)=−1,x>0,v(0,t)=0,t>0d1nÙ.u>¯)ª§x>t,ZZZx+ttx+(t−τ)112u(x,t)=(−1)dξ+dξdτ2x−t20x−(t−τ)(1+τ)3t=−1+t06x

5、τ)−x(1+τ)32t−xx−(t−τ)(1+τ)322x+x+4xt=−(1+t−x)(1+t)12c+4Ïdc>1§limx→+∞u(cx,x)=−2,00,(1)u(x,0)=sinx,uy(x,0)=x,−∞0,(2)u(0,y)=ux(0,y)=0,−∞

6、´dydy=−1,=3dxdxÙ)Ǒx+

7、y=C1,y−3x=C2.-ξ=x+y,η=y−3x.o§zǑuξη=0,ÙÏ)´u(x,y)=F(x+y)+G(y−3x)d^µ′′sinx=u(x,0)=F(x)+G(−3x),x=uy(x,0)=F(x)+G(−3x)S33é1§ü>'uxÈ©§:121x=F(x)−G(−3x)+C23ù±)3212G(−3x)=(2sinx−x+2C),F(x)=(2sinx+3x−6C)88¤±¯)´213yyu(x,y)==F(x+y)+G(y−3x)=sin(x+y)+sin(x−)++xy4433(2).¤§éAAƧ

8、´dy=−sinx±1dxÙÈ©Ǒy

9、=cosx+x+C1,y=cosx−x+C2.-ξ=cosx+x−y,η=cosx−x−y.o§zǑη−ξ−4uξη=eÙÏ)´1η−ξ1−2xu(ξ,η)=e+F(ξ)+G(η),½öu(x,y)=e+F(cosx+x−y)+G(cosx−x−y)44d^µ10=u(0,y)=+F(1−y)+G(1−y)41′′0=ux(0,y)=−+F(1−y)−G(1−y)2é1§ü>'uxÈ©§:1−F(1−y)+G(1−y)=−y+C2ù±)1yC1yCF(