Solved Midterm Exam - Partial Differential Equations | MTH 3326, Exams of Differential Equations

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- MATH 3326 MIDTERM #1 FALL SEMESTER 2008 Lance L. Littlejohn Name ___ SOLUTIONS Instructions: Show all work. Partial credit can only be given if sufficient work accompa- nies each answer. Calculators may be used but exact answers are required. This examination is out of 70 points. GOOD LUCK! Problem No. | Points 1. 2s 3. 4. 5. Grade /70 1. (7 Points) Let c > 0. By computing uz, tz, uz, and t,, show that 1 1 cet (co) u(a,t) = 5 (ple + et) + ola — et) + = vls}ds 2 2c Sst is a solution of the wave equation sey Un = C Urry where is an arbitrary twice differentiable function and 1 is a differentiable function. Then compute, and simplify, u(z,0) and u,(x,0) My b,th= SEP wre) —¥ (x-et)] +27 Ly xr etc ~y-ct)Cey] a) = 5 EP ier ck) =P bee) ak Eaten) ry ce] (nz) Uy, %AD= oty Mxset +" y- ct] at Lv 'Gcg¢et)—p (x-c8)] Uy t= tly "ixrct) ey" x-cb)] eel (xact)-y K-c#)) Myehvt= PLP" sect) +o" (x-ct)] +E LY he ect)-y xe] (3) Ae Ok t= ES Lovet Hplta-ct] +S Loy xect)-¥ tect) Comparing Q) Arecl (2), we have. Verified tat ult) soticfes (6) let teoan (uo): w&,o)< ELpieylode dt Pyisyds = Pix) Let t=0 4m U1): = ’ , = tn U)2 MW tx,a)= ety O)-9' er] +3 Dwi) HVio]) = yee