Student Exercises

#1

> int((2*x^2+3*x+1)*exp(x),x);

(2-x+2*x^2)*exp(x)

#2 a, b, c.

> dsolve(diff(y(x),x$2)+diff(y(x),x)-6*y(x)=3*exp(2*x)-exp(x),y(x));

y(x) = exp(-3*x)*_C2+exp(2*x)*_C1-3/25*exp(2*x)+1/4*exp(x)+3/5*x*exp(2*x)

With initial conditions included.

> dsolve({diff(y(x),x$2)+diff(y(x),x)-6*y(x)=3*exp(2*x)-exp(x),D(y)(0)=1,y(0)=2},y(x));

y(x) = 67/100*exp(-3*x)+27/25*exp(2*x)+1/4*exp(x)+3/5*x*exp(2*x)

> dsolve(diff(y(x),x$2)+4*diff(y(x),x)+4*y(x)=2*exp(3*x)+exp(-2*x),y(x));

y(x) = exp(-2*x)*_C2+exp(-2*x)*x*_C1+1/50*(4*exp(5*x)+25*x^2)*exp(-2*x)

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

y(x) = -2*x+x*exp(x)-exp(x)+exp(x)*_C1+_C2

#3 a, b

> dsolve(diff(y(x),x$2)-diff(y(x),x)-6*y(x)=x^3,y(x));

y(x) = exp(-2*x)*_C2+exp(3*x)*_C1+13/216-7/36*x+1/12*x^2-1/6*x^3

> dsolve(diff(y(x),x$2)-2*diff(y(x),x)=3*x^2+2*exp(2*x)+exp(x),y(x));

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

#4 a, b

> dsolve(diff(y(x),x$2)-diff(y(x),x)-6*y(x)=x*exp(x),y(x));

y(x) = exp(-2*x)*_C2+exp(3*x)*_C1+1/36*(-1-6*x)*exp(x)

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

y(x) = 1/24*(12*_C1-3+6*x-6*x^2+4*x^3)*exp(2*x)+_C2

#5 a, b

> dsolve(diff(y(x),x$2)-diff(y(x),x)-6*y(x)=exp(x)*sin(x),y(x));

y(x) = exp(-2*x)*_C2+exp(3*x)*_C1-1/50*exp(x)*(cos(x)+7*sin(x))

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

y(x) = exp(x)*sin(x)*_C2+exp(x)*cos(x)*_C1-1/2*exp(x)*(-sin(x)+cos(x)*x)