যেকোন গণিত সমস্যা সমাধান করুন

ধাপ ধাপে

1. - y{\left(x \right)} + \frac{d^{2}}{d x^{2}} y{\left(x \right)} = 0

   The differential equation.

2. \text{order } 2

   Order 2: the highest derivative present.

3. \text{Constant coefficients, homogeneous}

   Try y = e^{rx}: the characteristic polynomial in r gives the roots, and each root contributes a term to the general solution.

4. r^{2} - 1 = 0

   Characteristic equation of the homogeneous part (substitute y = e^{rx}).

5. r = -1,\ r = 1

   Its roots.

6. y{\left(x \right)} = C_{1} e^{- x} + C_{2} e^{x}

   General solution (C₁, C₂ … are arbitrary constants).

7. \checkmark

   Verified: substituting the solution back into the equation gives 0.

উত্তর প্রকাশ করুন

y{\left(x \right)} = C_{1} e^{- x} + C_{2} e^{x}