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# Question about second order differential equations watch

1. Say we have a second order ODE

ay'' + by' + cy = f(x) + g(x)

where f(x) and g(x) are functions of x.

And say we want to find a particular integral for this.

What if we find a particular integral to
ay'' + by' + cy = f(x)
and one to
ay'' + by' + cy = g(x)
?

If we add together these particular integrals will we get a particular integral for
ay'' + by' + cy = f(x) + g(x)
?
2. (Original post by gangsta316)
Say we have a second order ODE

ay'' + by' + cy = f(x) + g(x)

where f(x) and g(x) are functions of x.

And say we want to find a particular integral for this.

What if we find a particular integral to
ay'' + by' + cy = f(x)
and one to
ay'' + by' + cy = g(x)
?

If we add together these particular integrals will we get a particular integral for
ay'' + by' + cy = f(x) + g(x)
?
Write the differential operator as L. L is linear from the chain rule.

then if L(y)=f(x) and L(w)=g(x)

then L(y+w) = L(y) + L(w) = f(x) + g(x)

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