y%3De%5E2x%2Bx%5E2%2Bx%2Bsin2x.jpeg "(d^3+2d^2+d)y=e^2x+x^2+x+sin2x")
Solving the Differential Equation: (d^3 + 2d^2 + d)y = e^(2x) + x^2 + x + sin(2x)
This article will guide you through solving the given third-order linear non-homogeneous differential equation:
(d^3 + 2d^2 + d)y = e^(2x) + x^2 + x + sin(2x)
We will break down the solution process into distinct steps.
1. Finding the Complementary Solution (y_c)
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Step 1: Find the roots of the auxiliary equation:
The auxiliary equation is obtained by replacing the differential operators (d) with the variable (m) in the homogeneous part of the equation:
m^3 + 2m^2 + m = 0
Factoring out 'm' gives: m(m^2 + 2m + 1) = 0
Further factoring: m(m + 1)^2 = 0
This gives us roots: m = 0, m = -1 (repeated root)
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Step 2: Form the complementary solution (y_c)
For distinct roots, we use the term 'e^(mx)' in the solution. For repeated roots, we multiply the term by 'x'.
Therefore, the complementary solution is: y_c = c_1 + c_2e^(-x) + c_3xe^(-x)
2. Finding the Particular Solution (y_p)
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Step 1: Identify the form of the particular solution based on the non-homogeneous terms:
The non-homogeneous terms are: e^(2x) + x^2 + x + sin(2x)
We need to consider each term separately:
- e^(2x): Since '2' is not a root of the auxiliary equation, we assume a particular solution of the form Ae^(2x).
- x^2 + x: Since the auxiliary equation has a root of '0', we assume a particular solution of the form Bx^2 + Cx + D.
- sin(2x): We assume a particular solution of the form E sin(2x) + F cos(2x).
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Step 2: Combine the individual forms and adjust for repetition:
The general form of the particular solution is:
y_p = Ae^(2x) + Bx^2 + Cx + D + E sin(2x) + F cos(2x)
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Step 3: Determine the coefficients (A, B, C, D, E, F) by substituting y_p into the original equation:
Substitute y_p and its derivatives into the equation (d^3 + 2d^2 + d)y = e^(2x) + x^2 + x + sin(2x) and solve for the constants. This involves differentiation, substitution, and equating coefficients.
3. Combining the Solutions
The general solution (y) is the sum of the complementary solution (y_c) and the particular solution (y_p):
y = y_c + y_p
y = c_1 + c_2e^(-x) + c_3xe^(-x) + Ae^(2x) + Bx^2 + Cx + D + E sin(2x) + F cos(2x)
Conclusion
This process provides the general solution to the given differential equation. Remember that the constants (c_1, c_2, c_3, A, B, C, D, E, F) need to be determined using initial conditions or boundary values, if provided.