Math Problem Statement
y double prime + 4 yprime = x
Solution
The given differential equation is:
This is a linear non-homogeneous second-order differential equation. Let's solve it step by step.
1. Solve the Homogeneous Equation
The homogeneous part of the equation is:
Its characteristic equation is:
Factoring:
So, the roots are:
The general solution to the homogeneous equation is:
2. Solve for a Particular Solution
The non-homogeneous equation is:
We use the method of undetermined coefficients. Assume a particular solution of the form:
Substitute into the differential equation:
Equating coefficients of and constants:
- Coefficient of : → contradiction, adjust to . Let us refine the solution approach for a better guess for the particular solution since alone contradicts.
Now guess Yp -> Degree adjustmentsSince the right-hand side is , we revise our guess for the particular solution.
We let:
and substitute into the differential equation:
Step-by-Step Substitution:
-
First derivatives:
- ,
- .
-
Substitute into the equation:
-
Compare coefficients of and constants:
- Coefficient of : There is no match, which suggests the need to revise the guess for to a polynomial form.
Correct Guess: Quadratic Form
The minimal polynomial guess should be:
Now, compute derivatives:
- ,
- .
Substitute these into the equation:
Simplify:
Equating coefficients:
- Coefficient of : ,
- Constant term: ,
- No -term adjustment needed since no constant appears.
Particular Solution:
3. General Solution
Combine the homogeneous solution with the particular solution :
Let me know if you'd like detailed verification steps or further explanation!
5 Related Questions:
- How do we solve non-homogeneous differential equations using other methods, like variation of parameters?
- What is the role of the characteristic equation in solving differential equations?
- How do we select the correct form of the particular solution for various right-hand sides?
- What happens if the particular solution overlaps with the homogeneous solution?
- Can this method be applied to higher-order differential equations?
Tip:
Always check if the particular solution conflicts with the homogeneous solution and adjust your guess accordingly (e.g., by multiplying by ).
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Math Problem Analysis
Mathematical Concepts
Linear Differential Equations
Second-Order Differential Equations
Method of Undetermined Coefficients
Formulas
Characteristic equation: r^2 + 4r = 0
General solution: y = y_h + y_p
Particular solution guess: y_p = Ax^2 + Bx + C
Theorems
Superposition principle for linear differential equations
Method of Undetermined Coefficients
Suitable Grade Level
Undergraduate (Calculus II or Differential Equations)
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