Solve 13 and 14 Verify the functions form a fundamental set of solutions of the

Question

Solve 13 and 14

Verify the functions form a fundamental set of solutions of the DE on the interval. Form the general solution. x^2y" + xy' + y = 0, y_1 = cos(ln x), y_2 = sin(ln x), (0, infinity) (There are 3 steps to show a set of solutions is a fsos: 1. Make sure the set is large enough by observing that the number of solutions is the same as the order of the equation, 2. Verify that y_1 and y_2 are actually solutions and 3. Show they are linearly independent.) Verify that the given 2 parameter family of functions is the general solution of the nonhomogeneous DE on the interval. y" - y = sec x, y = c_1 cos x + c_2 sin x + x sin x + (cos x) ln(cos x), (-pi/2, pi/2) (First identify y_h and y_p in the general solution. Then there are two main steps 1. Show y_h is the general solution of the associated homogeneous equation. This requires identifying the homogeneous equation, y_h; and its associated fsos, then following the same steps as the previous problem and 2. Show y_p is a solution of the nonhomogeneous equation.)

Explanation / Answer

13. Given diff equn is :x^2y''+xy'+y=0

solutions are :y2=cos(lnx), y2=sin(lnx)

as order of diff equation =2

to check whether the two solutions are functional set of solutions of DE

1. order of DE =no. of solutions

as order=2

and no. of solutions =2

2. each function in the set must be a solution of the DE

as y1=cos(lnx)

so, y1'=-sin(lnx)/x

y1''=sin(lnx)-cos(lnx)/x2

y2=sin(lnx)

y2'=cos(lnx)/x

y2''=-(sinlnx+coslnx)/x2

as given DE is x2y''+xy'+y=0

for y1=cosx

x2*{sinlnx-coslnx/x2}-x{sinlnx/x}+cos(lnx)=0

so y1=cos(lnx) is a fsos

similarly y2=sinlnx is also fsos

now 3. function must be linearly independent

as from wronskion rule

w= 1 that is not equal to zero

so all the conditions are satisfying hence verified and hence

general solution will be

y=c1cosx+c2sinx

14. given de is y''+y=secx

to find general solution

y(x)=yc+Yp

first we will find homogeneous equation of DE which is y''+y=0

so from quadratic equation

r2=-1

r=i,-i

a=0,b=1

yc=c1cosx+c2sinx

now forYpw=1

so Yp=-cosx {sinx.secxdx+ sinx{cosx.secx dx

=cos lncosx+xsinx

so geeral solution is y(x)=yc+Yp

y(X)=c1cosx+c2sinx+cosln(cosx)+xsinx

also it is satisfyyiing all the three condition which I explained in question 13

hence verified

sorry for inconvinience, iI have the pic but it is not uploading. so I have written it in short time

Thanks

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