3y′′+y′=0A second order linear homogeneous The geveral solution isy(x)=yh3y′′+y′=0→3r2+r=0∴r(3r+1)=0→r=0,r=−31∴y(x)=c1e0+c2e−31x(1)⟹y(x)=c1+c2e−31xy′′′+14y′′+49y′=0 Alinear homogeneous solution The general solution y(x)=yhr3+14r2+49r=0∴r(r2+14r+49)=0∴r(r+7)2=0⇒r=0,r=−7,r=−7∴y(x)=c1e0+c2e−7x+c3xe−7xy(x)=c1+c2e−7x+c3xe−7xy(0)=0⟶c1+c2=0(1)y′(x)=−7c2e−7x+c3(−7xe−7x+e−7x)y′(0)=1∴1=−7c2+c3(0+1)→−72+c3=1(2)y′′(x)=49c2e−7x+c3(49xe−7x−7e−7x−7e−7x)y′′(x)=49c2e−7x+c3(49xe−7x−14e−7x)y′′(0)=−8∴−8=49c2+14C3(3) SoLv (1),(2),(3)∴c1=496,c2=−496,c3=71y(x)=496−496e−7x+71xe−7xy(x)=491e−7x(6e7x+7x−6)
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