The percent grаde will be recоrded. Existence-Uniqueness Theоrem: If f(x, y) аnd df/dy аre cоntinuous on a rectangle R in the xy-plane containing the initial condition y(x0)=y0, then the initial value problem y’=f(x,y), y(x0)=y0 has a unique solution in R. 6pts Determine whether the Existence-Uniqueness Theorem can be used to determine if the initial value problem: y’ = 1/x + y1/3, (1,1) has a unique solution. Please indicate the largest possible rectangle R from the Theorem. 21pts First order ODEs: Solve the following. Provide solutions in explicit form if possible. Theorem: M(x,y) dx + N(x,y) dy = 0 is an exact equation if dM/dy = dN/dx. a. (y4 + 1)cos x dx - y3 dy = 0 b. (12x – y)dx – 3x dy = 0 c. (x3 + y/x)dx + (y2 + ln x) dy = 0 6pts Homogeneous ODE: Solve y iv + 5y ‘’ – 36y = 0. 10pts Nonhomogeneous ODEs: Solve the following with either undetermined coefficients or variation of parameters to solve 3y ‘’ – y’ – 2y = 4x + 1, y(0) = 1 and y’(0) = 0 10pts Systems: Solve the following. x1’ = 2x1 – 4x2 x2’ = 2x1 – 2x2 15pts Solve the initial value problem for y(t) using the method of Laplace transforms. y ’’ + 4y’ + 3y = 1 y(0)=0, y’(0) = 0 Taylor polynomial about 0: pn(x) = f(0) + f’(0)x + f ‘’(0)/2! x2 + f ‘’’(0)/3! x3 + … + f (n)(0)/n! xn 7pts Determine the first three nonzero terms in the Taylor polynomial approximations for the given initial value problem y ’’ – 2y’ + y = 0; y(0)=0, y’(0) = 1 Theorem: Consider the differential equation A(x) y” + B(x) y’ + C(x) y = 0. If the functions p(x) = B(x)/A(x) and q(x) = C(x)/A(x) are analytic at x =0, then the general solution is produced by the power series centered at x=0: y(x) = a0 + a1x + a2 x2 + a3 x3 + … 10pts Determine the first four nonzero terms in the power series expansion about x=0 for a general solution in the given ODE y ’’ + xy’ + y = 0