Question

Show that if y₁and y₂ are solutions of the inhomogeneous ODE p(x)((d²y)/(dx²)) + q(x)((dy/dx)) + r(x)y = s(x), then y = y₁ − y₂ is a solution of the associated homogeneous ODE.

Answer 1

To show that y = y₁ - y₂ is a solution of the associated homogeneous ODE, substitute y into the inhomogeneous ODE and simplify the equation to verify that it becomes equal to zero.

Step-by-step explanation:

To show that y = y₁ - y₂ is a solution of the associated homogeneous ODE, we need to substitute y into the ODE and verify that it satisfies the equation.

Considering the inhomogeneous ODE: p(x)((d²y)/(dx²)) + q(x)((dy/dx)) + r(x)y = s(x), substitute y = y₁ - y₂ into the equation:

p(x)((d²(y₁ - y₂))/(dx²)) + q(x)((d(y₁ - y₂))/(dx)) + r(x)(y₁ - y₂) = s(x)

Expanding and simplifying the equation:

p(x)((d²y₁)/(dx²)) - p(x)((d²y₂)/(dx²)) + q(x)((dy₁)/(dx)) - q(x)((dy₂)/(dx)) + r(x)y₁ - r(x)y₂ = s(x)

Since y₁ and y₂ are solutions of the inhomogeneous ODE, the left side of the equation becomes:

s(x) - s(x) = 0

Therefore, y = y₁ - y₂ satisfies the associated homogeneous ODE.