Question

Difference between linear and nonlinear differential equation

Answer 1

Linear differential equations are those which can be reduced to the form Ly=fLy=f, where LL is some linear operator.

Your first case is indeed linear, since it can be written as:

(d2dx2−2)y=ln(x)(d2dx2−2)y=ln⁡(x)

While the second one is not. To see this first we regroup all yy to one side:

y(y′+1)=x−3y(y′+1)=x−3

then we simply notice that the operator y↦g(y)=y(y′+1)y↦g(y)=y(y′+1) is not linear (for example we can take two functions y1y1and y2y2 and notice that g(y1+y2)≠g(y1)+g(y2)g(y1+y2)≠g(y1)+g(y2)).

Answer 2

Linear differential equations are those which can be reduced to the form Ly=f, where L is some linear operator.

Your first case is indeed linear, since it can be written as:

(d2dx2−2)y=ln(x)

While the second one is not. To see this first we regroup all y to one side:

y(y′+1)=x−3

then we simply notice that the operator y↦g(y)=y(y′+1) is not linear (for example we can take two functions y1 and y2 and notice that g(y1+y2)≠g(y1)+g(y2)).