Working with Differential Equations

<br>y^3 - 2y^2 + y^1 = 2- 24e^(x) + 40e^(5x) ....(1)
<br>
<br>
<br>First set the right side equal to 0 and find the solution as in the
<br>case of a homogeneous equation (this is called complementary function
<br>y1) and then by trial find a particular solution (particular integral
<br>y2)
<br>
<br>Now the general solution is given by,
<br>
<br> Y = y1 + y2
<br>
<br>First evaluate y1
<br>................
<br>
<br>y^3 - 2y^2 + y^1 = 0
<br>
<br>ie, y (y^2 - 2y +1) = 0 This gives y = 0, +1, +1
<br>
<br>Now our y1 is given by
<br>
<br>y1 = C1 e^(0 x) + (C2 x+C3) e^(1 x)
<br>
<br>= C1 + (C2 x + C3) e^x ........(2)
<br>
<br>
<br>Now Find y2
<br>...........
<br>
<br>We will use the method of undetermined coefficients to find y2,
<br>
<br>Let y2 = Ko + C1 e^x + C2 e^5x ......(3)
<br>
<br>(If, for example, a term 4x were present in the original equation, we
<br>will have a K1 x term added with this and so on for higher powers of x)
<br>
<br>We will determine Ko, C1 and C2 in this by the above said method
<br>For this find the derivatives y2^3, y2^2, y2^1 and substitute them in
<br>our original equation.
<br>
<br>y2^1 = 0 + C1 e^x + 5 C2 e^5x
<br>
<br>y2^2 = C1 e^x + 25 C2 e^5x
<br>
<br>y2^3 = C1 e^x + 125 C2 e^5x
<br>
<br>Substitute back in (1)
<br>
<br>(C1 e^x + 125 C2 e^5x) -2(C1 e^x + 25 C2 e^5x) + C1 e^x + 5 C2 e^5x
<br>
<br> = 2- 24e^(x) + 40e^(5x)
<br>
<br>Simplify the left side and equate the coefficients of like terms and
<br>find C1, K0 and C2
<br>
<br>We get, Ko = 0, C1 = 0, C2 = 40/80 = 1/2
<br>
<br>Give these values to (3) to get our original y2
<br>
<br>y2 = 0 + 0 + (1/2) e^(5x)
<br>
<br>Now the general solution of the given differential equation is,
<br>
<br>Y = y1 + y2
<br>
<br> = C1 + (C2 x + C3) e^x + (1/2) e^(5x) Answer
<br>
<br>
<br>
<br>......................................................................

This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!