Question

Use the elimination method to find a general solution. x(t), y(t) for the given system. · = x + 2y dy = -4x - 3y dt

Answer 1

equation is in given system of differential ole - U+dy - -4*-zy -H) - (**) then given syorem can be written as ! (Driju - 2y so uni (D+3)y=0 - -6 so by why one by is known elimination any this method given syorum , ②. that's method wellr solve doom ez elimination o one as Now (D-1) x 2" (2) - 4x02 (1) gives ... 0+3)(0-1) + 8y = 0

X dla or to find so (8% + 2D + 5)% =0 - (ie they tady + 5y =o) yet), we nec to solve est ® auxiliary esuanion is not an tS to so its roots are: me -1 + 21 hence Solusion is ylt) = et (Alogat & B Binat) - a (whose A&B are arbitrary constanto Now voily this in esvasion (**) , we get orijinal -e (Aboat &B Binat) tet Asihat & 2B Cosat ) = run.Bê (Abort + B852) -Unlt) = 28 (Alooat & B sinat) + 28 (-Asinat & Boost) os miles € (2+28) Godt + (-34-2 ) sinat) -(0) tt (-2A + alt) = Fox 20+) = sell ( 147B) Godt + (+A+B) Biovt) +B) Coost & (-A+B) sinat) Helt ) = Hence with theo equacion t of B constants ① and aR if we arbitrary are gir ar rescisel sonision Constants and we can find given initial conditions.