(Solved): SECTION B 2. (a) Prove Abel's identity (see the formula sheet). (b) Find two linearly independent so ...

2. (a) Prove Abel's identity (see the formula sheet). (b) Find two linearly independent solutions of the equation \\[ \\frac{\\mathrm{d}^{2} y}{\\mathrm{~d} x^{2}}+4 \\frac{\\mathrm{d} y}{\\mathrm{~d} x}+4 y=0 \\] Verify by direct calculation that Abel's identity holds for these two solutions, and identify the corresponding value of the constant \\( C \\). (c) Write Abel's identity for the differential equation \\[ (x+1) y^{\\prime \\prime}+x y^{\\prime}-y=0 . \\] Verify by substitution that \\( y_{1}(x)=\\mathrm{e}^{-x} \\) is a solution of this equation. Use these two results, Abel's identity and the particular solution \\( y_{1}=\\mathrm{e}^{-x} \\), to find the general solution of this equation. (7)