Math Problem Statement
x^2y''+2xy'-42y=0, y(1)=-8,y'(1)=0
Solution
We are given the second-order linear differential equation:
with initial conditions:
Step 1: Solve the homogeneous differential equation
This is a Cauchy-Euler (or Euler-Cauchy) equation of the form: In our case, and .
The general solution to a Cauchy-Euler equation can be found by assuming a solution of the form: where is a constant. We differentiate with respect to :
Step 2: Substitute into the differential equation
Substitute , , and into the original equation: [ x^2 [r(r-1) x^{r-2}] + 2x [r x^{r-1}] - 42 x^r = 0. ] Simplify each term: Factor out : We are left with the characteristic equation: Simplify:
Step 3: Solve the characteristic equation
Solve the quadratic equation using the quadratic formula: The two roots are:
Step 4: General solution
The general solution to the differential equation is:
Step 5: Apply initial conditions
We now use the initial conditions to determine and .
- Initial condition 1:
Substitute into the general solution: Thus, we have:
- Initial condition 2:
First, differentiate : Substitute into the derivative: Thus, we have:
Step 6: Solve the system of equations
We now solve the system of equations:
From equation (2), solve for : Substitute this into equation (1):
Now, substitute into :
Final solution:
Thus, the solution to the differential equation is:
Would you like further details or explanations?
Related Questions:
- What is a Cauchy-Euler equation, and when is it used?
- How is the general solution for a Cauchy-Euler equation derived?
- Can the method used here apply to non-homogeneous equations?
- What would change if different initial conditions were given?
- How does the quadratic formula help in solving second-order differential equations?
Tip: To solve higher-order differential equations, always try to reduce them to familiar forms like Cauchy-Euler, which simplifies the solution process.
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Math Problem Analysis
Mathematical Concepts
Differential Equations
Cauchy-Euler Equations
Quadratic Equations
Formulas
Cauchy-Euler form: x^2y'' + axy' + by = 0
Characteristic equation: r^2 + (a-1)r + b = 0
Quadratic formula: r = (-b ± √(b^2 - 4ac)) / 2a
Theorems
Quadratic Formula
Cauchy-Euler Equation Solution Method
Suitable Grade Level
Undergraduate
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