Question

# In many physical applications, the nonhomogeneous term F(x) is specified by different formulas in different intervals of x. (a) Find a general solutio

Differential equations
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 $$\displaystyle{y}{''}+{y}=\left\{{x},{0}\leq{x}\leq{1},{1}\leq{x}\right\}$$ NoteNote that the solution is not differentiable at x = 1. (b) Find a particular solution of $$y''+y=\left\{x,0\leq x\leq 1;1,1\leq x\right\}$$ that satises the initial conditions y(0)=0 and y′(0)=1.
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)$$\displaystyle{y}={\left\lbrace{x},{0}\leq{x}\leq{1};{1},{x}\leq{1}\right.}$$