# Find a general solution of the equation y''+y={x,0\leq x\leq . 1,1\leq x

In many physical applications, the nonhomogeneous term F(x) is specified by different formulas in different intervals of x. (a) Find a general solution of the equation $$\displaystyle{y}{''}+{y}={\left\lbrace{x},{0}\leq{x}\leq.{1},{1}\leq{x}\right.}$$ NoteNote that the solution is not differentiable at x = 1. (b) Find a particular solution of y''+y= $$x,0\leq x\leq 1.1,1\leq x$$that satises the initial conditions y(0)=0 and y′(0)=1.

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delilnaT
a)$$\displaystyle{y}={\left\lbrace{c}{1}{\cos{{x}}}+{c}_{{{2}}}{\sin{{x}}}+{x},{0}\leq{x}\leq{1}.{c}_{{{1}}}{\cos{{x}}}+{c}_{{{2}}}{\sin{{x}}}+{1},{x}\geq{1}\right.}$$
b)$$\displaystyle{y}={\left\lbrace{x},{0}\leq{x}\leq{1.1},{x}\leq{1}\right.}$$