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2 PROBLEM 1 We will solve an initial value problem of the ODE y” + 4y = .x sin 2x. (a) Find the general solution y, for the homogeneous equation y” + 4y = 0.. Check, using Wronskian, that your solutiois are linearly independent. (b) Find a particular solution for the original nonhomogeneous equation. (c) Write down the general solution for the nonhomogeneous equation. (d) Solve the initial value problem for the above ODE with initial conditions y(O) 4 and y’(O) = 2. (‘i) 9t L( ( ) (() 0 -. 2XkX Cf 2 2 (k-) 2 (K÷I?) - S2 t’ - - (A x) ‘i ‘- ‘ti 2 ( A ‘+ (2i* 4 t 2 (C & 2)C PPE —) XS /- 2 ‘2- I C?(+ (C) q 11 ‘(Q) 2 (2 x •fr )5 h / (c 0 )c) ± (c C> 0 2 ( “1:4 [3’ 3 PROBLEM 2 Solve the following ODE via power series method. y’ + y x + (a) Explain that 0 is an ordinary (regular) point. That is, the power series method applies to this ODE. (b) Write down the recurrence relation for the coefficients. (c) Since this is an ODE of order 1, how many arbitrary constants ç are there in the general solution? (d) Compute the first 4 coefficients in terms of the arbitrary con stant(s). c ‘ (4) -rka c.c’.e6 f ô 9(x) 7 i4tI Pb! ?(IX Z. v-e k’c’4 / (1’) / pa P2’l XhI : (iii) RfrF c) (of) fllb L — (ni-i-i) x %, (s+f,1t 2 LcQffof C / ‘h c:j / <L z qJ: a / (—1 1 4 PROBLEM 3 Solve the following equation by the method of the characteristic curves. 6u 0u 8x 0y Verify your solution (by plugging it back to the PDE.) F ) Pt C ° 1 - -( ) pCSSih / 1 1 ( C) / (J) C’ n In / p