2014年下学期数学实验与数学建模作业习题7

2014年下学期数学实验与数学建模作业习题7

1.求下列微分方程的通解 （1）

'yxdyy?x? （2）y?cos? （3）（xcosy+sin2y）y`=1

xydxy?x(4)y???3y??y?excos2x (5) y''?3y'?y?excos2x

【1】求解命令： syms x y

y1=dsolve('Dy-(y+x)/(y-x)=0','x') y2=dsolve('Dy-cos(y/x)-x/y=0','x') y3=dsolve('Dy*(x*cos(y)+sin(2*y))=1','x') y4=dsolve('D2y+3*Dy-y=exp(x)*cos(2*x)','x') y5=dsolve('D2y+3*Dy-y=exp(x*cos(2*x))','x')

【2】运行结果： y1 =

x + 2^(1/2)*(x^2 + C15)^(1/2) x - 2^(1/2)*(x^2 + C15)^(1/2)

y2 =x*z%有问题 y3 =

-asin(x/2 + lambertw(0, -(C28*exp(- x/2 - 1))/2) + 1)

y4 =

C12*exp(x*(13^(1/2)/2 - 3/2)) + C13*exp(-x*(13^(1/2)/2 + 3/2)) + (13^(1/2)*exp(x*(13^(1/2)/2-3/2))*exp((5*x)/2-(13^(1/2)*x)/2)*(2*sin(2*x)

- cos(2*x)*(13^(1/2)/2 - 5/2)))/(13*((13^(1/2)/2 - 5/2)^2 + 4)) -

(13^(1/2)*exp(-x*(13^(1/2)/2+3/2))*exp((5*x)/2+(13^(1/2)*x)/2)*(2*sin(2*x)+ cos(2*x)*(13^(1/2)/2 + 5/2)))/(13*((13^(1/2)/2 + 5/2)^2 + 4)) y5 =

exp(x*(13^(1/2)/2 -3/2))*int((13^(1/2)*exp(x*cos(2*x))*exp(-(x*(13^(1/2)- 3))/2))/13, x, IgnoreAnalyticConstraints) + exp(-x*(13^(1/2)/2 + 3/2))*int(-(13^(1/2)*exp(x*cos(2*x))*exp((x*(13^(1/2) + 3))/2))/13, x, IgnoreAnalyticConstraints) + C30*exp(x*(13^(1/2)/2 - 3/2)) + C31*exp(-x*(13^(1/2)/2 + 3/2))

2．求下列初值问题的解

?d2xdx2?2n?x?0222dy?2?a2?2xy??(?2xy?)?0dtyyxxdx (2)?dt(1) ? ??dx??x?x0,yx?1?1?V0?t?0?dtt?0?【1】求解代码： syms x y a v0 t n x0

x1=dsolve('x*x+2*x*y-y*y+(y*y+2*x*y-x*x)*Dy=0','y(1)=1','x') x2=dsolve('D2x+2*n*Dx+a^(2)*x=0','x(0)=x0','Dx(0)=v0','t')

【2】运行结果： x1 =

(x*((- 4*x^2 + 4*x + 1)/x^2)^(1/2))/2 + 1/2 x2 =

(exp(-t*(n - (-(a + n)*(a - n))^(1/2)))*(v0 + n*x0 + x0*(-(a + n)*(a - n))^(1/2)))/(2*(-(a + n)*(a - n))^(1/2)) - (exp(-t*(n + (-(a + n)*(a -

n))^(1/2)))*(v0 + n*x0 - x0*(-(a + n)*(a - n))^(1/2)))/(2*(-(a + n)*(a - n))^(1/2))

?dxt?dt?5x?y?e3．求微分方程组?的通解.

dy2t??x?3y?e?dt【1】求解代码： syms x y a v0 t n x0

[x,y]=dsolve('Dx+5*x+y=exp(t)','Dy-x-3*y=exp(2*t)','t')

【2】运行结果: x =

-exp(-t*(15^(1/2) + 1))*(4*C42 - 4*exp(2*t + 15^(1/2)*t)*((7*exp(t))/12 - 15^(1/2)/165 - (3*15^(1/2)*exp(t))/20 + 1/22) + 15^(1/2)*(C42 - exp(2*t + 15^(1/2)*t)*((7*exp(t))/12 - 15^(1/2)/165 - (3*15^(1/2)*exp(t))/20 + 1/22)) + 4*exp(t*(15^(1/2) - 1))*exp(t*(15^(1/2) + 1))*(C41 - exp(2*t - 15^(1/2)*t)*((7*exp(t))/12 + 15^(1/2)/165 + (3*15^(1/2)*exp(t))/20 + 1/22)) - 15^(1/2)*exp(t*(15^(1/2) - 1))*exp(t*(15^(1/2) + 1))*(C41 - exp(2*t - 15^(1/2)*t)*((7*exp(t))/12 + 15^(1/2)/165 + (3*15^(1/2)*exp(t))/20 + 1/22))) y =

exp(-t*(15^(1/2) + 1))*(C42 - exp(2*t + 15^(1/2)*t)*((7*exp(t))/12 - 15^(1/2)/165 -

(3*15^(1/2)*exp(t))/20 + 1/22) + exp(t*(15^(1/2) - 1))*exp(t*(15^(1/2) + 1))*(C41 - exp(2*t - 15^(1/2)*t)*((7*exp(t))/12 + 15^(1/2)/165 + (3*15^(1/2)*exp(t))/20 + 1/22)))

4．求下列初值问题的解

?dx?dt?y?1?x(0)??2的特解。

(1)方程组?满足?dy?y(0)?0??x?1?dt?dxt?x?2y?e?(2)微分方程组?dt在初始条件xt?0?1,yt?0?0下的特解.

dy??x?y?0?dt?dx?dt?3x?y?0,xt?0?1(3)微分方程组?的特解.

dy??8x?y?0,yt?0?4?dt【1】求解命令： syms x y t

[x,y]=dsolve('Dx=y+1','Dy=x+1','x(0)=-2','y(0)=0','t')

[x,y]=dsolve('Dx+x+2*y=exp(t)','Dy-x-y=0','x(0)=1','y(0)=0','t') [x,y]=dsolve('Dx+3*x-y=0','Dy-8*x+y','x(0)=1','y(0)=4','t')

【2】运行结果：