Mock exam for Higher Order Ordinary Linear Differential Equation (MUC and MVP), Exams of Engineering Mathematics

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MUC without yp duplication y+ 2y’ + Sy = e* sinx Solution: Solving for yc Aux Eqn: m+2m+5=0 m?+2m+1=-4 (m+ 1)? =-4 m=-1+2i Ve = e-*(C, cos 2x + Cz sin 2x) Ve = Cye™* cos 2x + Czpe™* sin 2x Solving for yp f(x) = e* sinx Transformation for yp Vp = (Ae*)(B cos x + Csin x) Vp = ABe* cosx + ACe* sinx Yp = Me*cosx + Ne*sinx M=AB & N=AC Vp = Me* cosx — Me* sinx + Ne* sinx + Ne* cosx Yn = (M + N)e* cosx + (N — M)e*sinx Yp = (M + N)e* cosx — (M + N)e*sinx + (N — M)e* sinx + (N — M)e* cosx Yp = 2Ne* cosx — 2Me* sinx Vp t+ 2Ypy + S¥py = e* sinx (2Ne* cosx — 2Me* sinx) + 2[(M + N)e* cosx + (N — M)e* sin x] + 5(Me* cos x + Ne* sinx) = e* sinx (2N + 2M + 2N + 5M)e* cosx + (—2M + 2N — 2M + SN)e* sinx = e* sinx (AN + 7M)e* cosx + (7N — 4M)e* sinx = e* sinx 7 e* cosx: 4N+7M=0 > N=- 7M 7 e* sinx: 7N-4M=1 + 7(-1M)-4M=1 5 y — 4 4 M=—os 7 N=e 4 oxcosy seers =—-—e cosx+—e* sinx ye 65 65 Solving for y 7 V=Vet+Vp = Cye* cos 2x + C,e* sin 2x raid cosx+ ose sinx Find v's: 1 cosx sinx 0 —sinx cosx 0 -cosx —sinx. |A| = (sin? x) + cos? x =1 Solve 0 cos x sinx 0 —sinx cosx tanx+3 . ———_ -cosx —sinx 2: tanx +3 tanx +3 tanx +3 |Ay| = E>) (cos # x) — E>) (- sin? x) = >) (cos? x + sin? x) _tanx+3 2 tanx +3 yd _ tanx+3 Al 1 _ tanx+3 | tanx | Sax — Insecx | 3 my = | —z— dx= tox 1 0 sinx 0 0 cos x tanx +3 . 2 sin x sl (mats) ) —cosxtanx—3cosx —sinx—3cosx = —cosx) = = 2 2 eos 2 2 —sinx — 3cosx . ,_ a2] 72 __ ~sinx—3cosx 72 = Tal 1 = 2 _ ee -(> [=e — 08% | —3.sin# Kan 2 2 2g 2 1 cosx 0 0 —-sinx 0 tanx +3 0 -cosx ——>— 2 —sin? x tanx +3 . —sinxtanx—3sinx —Gogqx_ 7 3sinx _ asin? x 3sinx —(1—cos? x) 3sinx ~ 2cosx 20 2cosx 2 _ ot cos?x 3sinx ~ 2cosx | 2cosx 2 —secx cosx 3sinx a ar a) =secx , cosx 3sinx ot = Bal 8 2 7 2 ——seex | cosx’ 3sinx 3 Al 1 2 2 2 _ —— sen) a -(—Sa + COSH +f 3sinx | 3 = 2 2 ae eee ew z z _ —In|tanx + secx| LA Rd 7 2 2 2 ]Solution for yp using MVP: Yp = V1 + v2 cosx + v3 sinx (asx 3 ) + (ee See = sx 2 2 2 2 (lene seca sinx | 3cosx 2 2 2 Insecx 3 1 2 a) sinxIn|tanx + secx +5xt75 (cos x + sin’ a a ae) Insecx 3 1 sinxIn|tanx + secx| wg t9%t2 2 ) cos x ) sinx Complete Solution: Insecx 3 1 sinx In| tanx + secx | 2 12**2 2 y=C,+C,cosx+C3sinx+ Engr. Rodrigo L. Cardiz connected a resistor (R = 3 @), inductor (L = ; H),and capacitor (€ = 0.08 F) in series to an applied voltage E = 10 V. Assuming no initial current and charge when the voltage is first applied, find the subsequent charge (g) in the circuit. E=10v F Tle) = glo) =o