Topic 7.7 Finding Particular Solutions Using Initial Conditions and Separation of Variables

Question

#20: Find the particular solution of the differential equation using the initial conditions.

(dy)/(dx) = ysinx, y=e if x=0

Answer 1

dy/dx = ysinx

dy/y = sinx dx

∫dy/y = ∫sinx dx

ln(y) = -cos(x) + C₁

e^ln(y) = e^{-cos(x) + C1}

y = e^{-cos(x) + C1}

y = e^-cos(x) • e^C1 then let C₂ = e^C1 meaning:

y = C₂e^-cos(x)

Since we know that y=e if x=0, we can solve for C₂:

e = C₂e^-cos(0)

e = C₂e^-1

e = C₂/e

C₂ = e²

Therefore y = y = e²•e^-cos(x) or y = e^2-cos(x)