For each of the following differential equations find yc and

For each of the following differential equations, find y_c and y_p, using any (non-calculator) method you wish. Then, write the general solution. Y\'\' + y = x^2 y\'\' + 4y = sin(2x)

Solution

a.

First we solve homogeneous ode and get y_c

Homogeneous ode is

y\'\'+y=0

y\'\'=-y

Hence,

yc=A sin(x)+B cos(x)

Now based on the inhomogeneous part the guess for particular solution is

yp=Cx^2+Dx+E

yp\'=2Cx+D

yp\'\'=2C

Substituting gives

2C+Cx^2+Dx+E=x^2

COmparing coefficients gives

D=0,C=1,2C+E=0

E=-1/2

yp=x^2-1/2

So general solution is

y=A sin(x)+B cos(x)+x^2-1/2

b.

y\'\'+4y=sin(2x)

First we look for yc ie solution to homogeneous ode

y\'\'=-4y=-2^2y

Hence,

yc=A sin(2x)+B cos(2x)

Now we guess yp based on inhomogeneous part. Normally the guess would be

C sin(2x)+D cos(2x)

But this is already solution to homogeneous ode

So we take the guess

yp=x(C sin(2x)+D cos(2x))=xyc

yp\'=xyc\'+yc

yp\'\'=2yc\'+xyc\'\'

Substituting and using yc is a solution to homogeneous ode gives

2yc\'=sin(2x)

2(2C cos(2x)-2D sin(2x))=sin(2x)

This gives, C=0,D=-1/4

So,yp=-cos(2x)/4

Hence general solution is

y=A sin(2x)+B cos(2x)-cos(2x)/4