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Question: a draw characteristics and find the general solution to each...

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a. Draw characteristics and find the general solution to each of the following equations 2m+311x = 0; +111x = 0; u,-tux = 0; u, + ,214x = 0; Ixux0; (10) (x2 + 1)u, +14x=0; (x2-1)u, + ux = 0. (12) (13) b. Consider IVP problem ul,to f(x) as-oo < x < oo; does solution always exists? If not, what conditions should satisfy f(x)? Consider separately t>0 andt<0 Where this solution is uniquely determined? Consider separately t>0 andr<0 c. d. Consider this equation in t >0, x>0 with the initial conditionulo-fx) asx > O; where this solution defined? Is it defined everywhere in [0,x >0) or do we need to impose condition atx 0? In the latter case impose condition ulxsO = g(,) (I> 0) and solve this IVBP: e. Consider this equation in t >0,x <0) with the initial conditionu-fx) asx <O; where this solution defined? Is it defined everywhere in [ >0, x <0) or do we need to impose condition at x0? In the latter case impose conditionuo g() (> 0) and solve this IVBP f. Consider problems (d) ast < 0 g, consider problems (e) as , < 0;
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