1. Solve the given initial value problem. y''- 12y" +52y' - 96y=0; y(0) =1. y'(0)=0, y"(0)=0 y(1)= 2. Find a particular solution to the differential equation using the Method of Undetermined Coefficients. 7. Use the elimination method to find a general solution for the given linear system, wh differentiation is with respe " + 5y' + 10y =510e"cos 6: x' = 7x - 10y + sint y' = 5x - 3y - cost A solution is yo (!) = Eliminate x and solve the remaining differential equation for y. Choose the correct answer below. 3. Find a general solution to the differential equation. y" - 10y' + 25y=1-3 , 5 OA. y(1)= C, e 2+ Cate 2 + 70 cost+ 70 sint The general solution is y(!) = O B. y(1) = Cy e # cos 51 + Cell sin 5t+ an cost+ on sir Find a general solution to the differential equation using the method of variation of parameters. O c. y(1)= Ce "# cos 5t + Cze "2 sin 5t+ #7 co Sit + 40 Cost+ 40 Sint y" + 9y = 5 csc 3t OD. y(1)=C, e -2 + Cate -21 + 41 Co 21 + 70 cost+ 70 sint The general solution is y() = O E. The system is degenerate. 5. Solve the initial value problem below for the Cauchy-Euler equation. Now find x(1) so that x(1) and the solution for y(1) found in the previous step are a general solution to the system of differential equations. Select the correct choice below and, if necessary, fill in the answer box to complete your chol ry"()+ 10ty'(t) + 18y(t) = 0: y(1) = 2, y'(1) = 3 The solution is y()= O A. (!) B. The system is degenerate. 6. Find the general solutions to the following non-homogeneous Cauchy-Euler equation using variation of parameters 12"+tz' + 16z= = tan (4 In!) 8. Solve the given initial value problem. z(1)= =3x+y-e: x(0) =2 (Use parentheses to clearly denote the argument of each function.) 21 =4x + 3y: y(0) = -5 The solution is x(!) = and y() = Find at least the first four nonzero terms in a power series expansion about x = 0 for a general solution to the given differential equation. " + (x-2)y' +y=0 (x)= Type an expression in terms of a, and a, that includes all terms up to order 3.) 10. Find the first four nonzero terms in a power series expansion about x = 0 for the solution to the given initial value ; y" + (x - 2)y' - y = 0; y(0) = - 5, y'(0) =0 (x)= (Type an expression that includes all terms up to order 4.) Use the elimination method to find a general solution for the given linear system, when lifferentiation is with respec * = 7x - 10y+ sint y' = 5x - 3y - cost Eliminate x and solve the remaining differential equation for y. Choose the correct answer below. O A y(1)=Cy e 2 + Cate 2 + 20 cost* # sint 11. Find the first four nonzero terms in a power series expansion of the solution to the given initial value problem. O B. y(1) = C, e # cos 5t + Cze 2 sin 51 + 12 c St + 20 cost+ an sint by" - (cos xly' -y=0: y z -4. > (2) -4 O c. y(1) = Ce *21 cos 51 + Cze "2 sin St+ 70 cost+ 40 sint y(x)= (Type an expression that includes all terms up to order 4. Type an exact answer, using x as needed.) O D. y(!) =C1 e -21 + Cate " 21 4 71 CO *70 Sint O E. The system is degenerate. Now find x(1) so that x(1) and the solution for y(!) found in the previous step are a general solution to the system of differential equations. Select the correct choice below and, if necessary, fill in the answer box to complete your chok