CHAPTER 16 - Differential Equations

Section 16.2, page 10 (supplement)

Problem 20

We use the integrating factor as shown:

>   M:=x+2*y; N:=-x; rho:=1/x^3;

M := x+2*y

N := -x

rho := 1/(x^3)

>   diff(rho*M,y)-diff(rho*N,x);

0

So the integrating factor works. Now:

>   int(rho*M,x)+c(y); int(rho*N,y)+k(x);

-y/x^2-1/x+c(y)

-y/x^2+k(x)

These agree if c(y) is actually a constant and k(x)=-1/x+ a constant. So the general solution of the equation is

>   %%-c(y)=K;

-y/x^2-1/x = K

for any constant K. Or we could solve this for y and get:

>   y=solve(%,y);

y = -(1+K*x)*x

Section 16.3, page 16 (supplement)

Problem 10

>   simplify(dsolve({x*diff(y(x),x)+2*y(x)=x^3,y(2)=1},y(x)));

y(x) = 1/5*(x^5-12)/x^2

Section 16.4, page 21 (supplement)

Problem 6

>   dsolve(diff(y(x),x$2)+6*diff(y(x),x)+9*y(x)=0,y(x));

y(x) = _C1*exp(-3*x)+_C2*exp(-3*x)*x

Problem 18

>   dsolve({4*diff(y(x),x$2)-2*diff(y(x),x)+y(x)=0,y(0)=4,D(y)(0)=2},y(x));

y(x) = 4/3*3^(1/2)*exp(1/4*x)*sin(1/4*3^(1/2)*x)+4*exp(1/4*x)*cos(1/4*3^(1/2)*x)

Section 16.5, page 34 (supplement)

Problem 14

>   dsolve(diff(y(x),x$2)-3*diff(y(x),x)-10*y(x)=2*x-3,y(x));

y(x) = exp(-2*x)*_C2+exp(5*x)*_C1+9/25-1/5*x

Problem 43

>   dsolve(diff(y(x),x$2)+2*diff(y(x),x)=x^2-exp(x),y(x));

y(x) = 1/6*x^3-1/4*x^2-1/3*exp(x)-1/2*exp(-2*x)*_C1+1/4*x+_C2

Problem 58

>   dsolve({x*diff(y(x),x$3)-2*diff(y(x),x$2)=0,y(1)=-5,D(y)(1)=2,D(D(y))(1)=3},y(x));

y(x) = -25/4+1/4*x^4+x