Given a solution y_1 of a differential equation y\" + P(x) y\' + Q(x) y=0 the re

Question

Given a solution y_1 of a differential equation y" + P(x) y' + Q(x) y=0 the reduction formula y2=y_1 (x) integral e^-f p(x) dx/y^2_1 (x) dx Gives another solution y2 which is linearly independent of y1, Use this formula to compute a solution of the equation 9y"- 12y' + 4y=0 that is linearly independent of y1=e^2 pi/3. Knowing that y=c1 e^x + c_2 e^-x is the general solution of the equation y' - y = 0 find the solution of the initial value problem y' - y=0, y(0) =1, y'(0)=-1. Use the method of coefficients to find the general solution of equation y' - 9y = 3e^3x. Find the general solution of the second order differential equation. Y^n - y' + y=0.

Explanation / Answer

( d^2 y(x))/( dx^2) - 9 y(x) = 3 e^(3 x):

The general solution will be the sum of the complementary solution and particular solution.

Find the complementary solution by solving ( d^2 y(x))/( dx^2) - 9 y(x) = 0:

Assume a solution will be proportional to e^( x) for some constant .

Substitute y(x) = e^( x) into the differential equation:

( d^2 )/( dx^2)(e^( x)) - 9 e^( x) = 0

Substitute ( d^2 )/( dx^2)(e^( x)) = ^2 e^( x):

^2 e^( x) - 9 e^( x) = 0

Factor out e^( x):

(^2 - 9) e^( x) = 0

Since e^( x) !=0 for any finite , the zeros must come from the polynomial:

^2 - 9 = 0
Factor:

( - 3) ( + 3) = 0

Solve for :

= -3 or = 3

The root = -3 gives y1(x) = c1 e^(-3 x) as a solution, where c1 is an arbitrary constant.

The root = 3 gives y2(x) = c2 e^(3 x) as a solution, where c2 is an arbitrary constant.

The general solution is the sum of the above solutions:

y(x) = y1(x) + y2(x) = c1 e^(-3 x) + c2 e^(3 x)

Determine the particular solution to ( d^2 y(x))/( dx^2) - 9 y(x) = 3 e^(3 x) by the method of undetermined coefficients:

The particular solution to ( d^2 y(x))/( dx^2) - 9 y(x) = 3 e^(3 x) is of the form:

yp(x) = x (a1 e^(3 x)), where a1 e^(3 x) was multiplied by x to account for e^(3 x) in the complementary solution.

Solve for the unknown constant a1:

Compute ( d^2 yp(x))/( dx^2):

( d^2 yp(x))/( dx^2) = ( d^2 )/( dx^2)(a1 e^(3 x) x)

= a1 (6 e^(3 x) + 9 e^(3 x) x)

Substitute the particular solution yp(x) into the differential equation:

( d^2 yp(x))/( dx^2) - 9 yp(x) = 3 e^(3 x)

a1 (6 e^(3 x) + 9 e^(3 x) x) - 9 (a1 e^(3 x) x) = 3 e^(3 x)

Simplify:

6 a1 e^(3 x) = 3 e^(3 x)

Equate the coefficients of e^(3 x) on both sides of the equation:

6 a1 = 3

Solve the equation:

a1 = 1/2

Substitute a1 into yp(x) = a1 e^(3 x) x:

yp(x) = 1/2 e^(3 x) x

The general solution is:

Answer:

y(x) = yc(x) + yp(x) = 1/2 e^(3 x) x + c1 e^(-3 x) + c2 e^(3 x)

IF U HAVE ANY DOUBT, PLZ ASK .N DNT FORGET TO LIKE MY ANSWER.

Related Questions

Navigate

• Browse (All)
• Browse G
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.