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Question Number 12245 by tawa last updated on 16/Apr/17
Solvethedifferentialequationdydx=4x+2y−38x−4y+5
Answered by mrW1 last updated on 17/Apr/17
dydx=4x+2y−38x−4y+5lety=u−pletx=v−qwithp,q=constantsdy=dudx=dv4(v−q)+2(u−p)−3=4v+2u+(−4q−2p−3)8(v−q)−4(u−p)+5=8v−4u+(−8q+4p+5)−4q−2p−3=0(i)−8q+4p+5=0(ii)(ii)−(i)×2:8p+11=0⇒p=−118(ii)+(i)×2:−16q−1=0⇒q=−116dydx=4x+2y−38x−4y+5⇒dudv=4v+2u8v−4uletu=wvdudv=w+vdwdvdudv=4v+2u8v−4u⇒w+vdwdv=4+2w8−4w=2+w4−2wvdwdv=2+w4−2w−w=2−3w+2w24−2w2−w2w2−3w+2dw=12vdv∫2−w2w2−3w+2dw=∫12vdv2∫12w2−3w+2dw−∫w2w2−3w+2dw=12lnv+C4×2×2−9=7∫12w2−3w+2dw=27tan−1(4w−37)∫w2w2−3w+2dw=14ln(2w2−3w+2)+327tan−1(4w−37)527tan−1(4w−37)−14ln(2w2−3w+2)=12lnv+C57tan−1(4w−37)−12ln(2w2−3w+2)=lnv+C57tan−1(4u−3v7v)−12ln[2(uv)2−3(uv)+2]=lnv+C57tan−1[4(y−118)−3(x−116)7(x−116)]−12ln[2(y−118x−116)2−3(y−118x−116)+2]=ln(x−116)+C57tan−1[4(16y−22)−3(16x−1)7(16x−1)]−12ln[2(16y−2216x−1)2−3(16y−2216x−1)+2]=ln(16x−116)+C57tan−1[64y−48x−857(16x−1)]−12ln[8(8y−1116x−1)2−6(8y−1116x−1)+2]=ln(16x−116)+C
Commented by tawa last updated on 17/Apr/17
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