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Differential-Equations · Differential-Equations

Y' + y = sin(x)

y{\left(x \right)} + \frac{d}{d x} y{\left(x \right)} = \sin{\left(x \right)}

Step by step

  1. y{\left(x \right)} + \frac{d}{d x} y{\left(x \right)} = \sin{\left(x \right)}

    The differential equation.

  2. \text{order } 1

    Order 1: the highest derivative present.

  3. \text{Exact}

    M dx + N dy = 0 with ∂M/∂y = ∂N/∂x: find the potential F with F_x = M, F_y = N; the solution is F = C.

  4. r + 1 = 0

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

  5. r = -1

    Its roots.

  6. y{\left(x \right)} = C_{1} e^{- x} - \frac{\sqrt{2} \cos{\left(x + \frac{\pi}{4} \right)}}{2}

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

  7. \checkmark

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

Reveal the answer
y{\left(x \right)} = C_{1} e^{- x} - \frac{\sqrt{2} \cos{\left(x + \frac{\pi}{4} \right)}}{2}