Math Problem Statement
solve
Solution
To solve the nonhomogeneous equation using the variation of parameters method, we proceed as follows:
Step 1: Solve the corresponding homogeneous equation
The homogeneous equation is:
This is a second-order differential equation with constant coefficients. Assume a solution of the form . Substituting into the homogeneous equation gives:
Solve for :
Thus, the solution to the homogeneous equation is: where and are arbitrary constants.
Step 2: Formulate the variation of parameters solution
The particular solution is given by: where and are functions to be determined.
Step 3: Derive the equations for and
The method of variation of parameters requires:
Step 4: Solve for and
The system of equations can be expressed as:
\cos(3x) & \sin(3x) \\ -3\sin(3x) & 3\cos(3x) \end{bmatrix} \begin{bmatrix} u_1'(x) \\ u_2'(x) \end{bmatrix} = \begin{bmatrix} 0 \\ 3\sec(3x) \end{bmatrix}.$$ Compute the determinant of the coefficient matrix: $$\Delta = \cos(3x)(3\cos(3x)) - (-3\sin(3x))\sin(3x) = 3(\cos^2(3x) + \sin^2(3x)) = 3.$$ Thus, $$\Delta = 3$$. The solutions for $$u_1'(x)$$ and $$u_2'(x)$$ are: $$u_1'(x) = \frac{1}{\Delta} \begin{vmatrix} 0 & \sin(3x) \\ 3\sec(3x) & 3\cos(3x) \end{vmatrix} = \frac{1}{3} \left( 0 - 3\sec(3x)\sin(3x) \right) = -\sec(3x)\sin(3x),$$ $$u_2'(x) = \frac{1}{\Delta} \begin{vmatrix} \cos(3x) & 0 \\ -3\sin(3x) & 3\sec(3x) \end{vmatrix} = \frac{1}{3} \left( 3\sec(3x)\cos(3x) - 0 \right) = \sec(3x)\cos(3x).$$ --- ### Step 5: Integrate to find $$u_1(x)$$ and $$u_2(x)$$ For $$u_1(x)$$: $$u_1(x) = \int -\sec(3x)\sin(3x) \, dx.$$ Let $$u = 3x$$, then $$du = 3dx$$. The integral becomes: $$u_1(x) = -\frac{1}{3} \int \sec(u)\sin(u) \, du = -\frac{1}{3} \ln|\sec(u) + \tan(u)| + C_1.$$ For $$u_2(x)$$: $$u_2(x) = \int \sec(3x)\cos(3x) \, dx.$$ Let $$u = 3x$$, then $$du = 3dx$$. The integral becomes: $$u_2(x) = \frac{1}{3} \int \sec(u)\cos(u) \, du = \frac{1}{3} \ln|\sec(u) + \tan(u)| + C_2.$$ --- ### Step 6: Write the general solution The general solution is: $$y = y_h + y_p = c_1 \cos(3x) + c_2 \sin(3x) + \left( u_1(x) \cos(3x) + u_2(x) \sin(3x) \right).$$ Simplify $$u_1(x)$$ and $$u_2(x)$$ to obtain the full solution. --- Would you like a detailed expansion of the integration steps or the final simplified expression? Let me know! --- ### Related Questions: 1. What is the general method to find the complementary solution for higher-order linear differential equations? 2. How does the variation of parameters method differ from undetermined coefficients? 3. Can variation of parameters be used for non-constant coefficient differential equations? 4. What are some limitations of the variation of parameters method? 5. How do trigonometric identities help simplify solutions involving secant and tangent? --- ### Tip: Always verify your general solution by substituting it back into the original differential equation to check correctness.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Differential Equations
Variation of Parameters Method
Trigonometric Functions
Formulas
General solution of nonhomogeneous equations: y = y_h + y_p
Variation of parameters: y_p = u_1(x)y_1(x) + u_2(x)y_2(x)
Theorems
Superposition principle for linear differential equations
Determinant method for solving systems of equations
Suitable Grade Level
Undergraduate Mathematics
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