Corresponding homogeneous differential equation
2y′′−5y′=0 Characteristic (auxiliary) equation
2r2−5r=0
r1=0,r2=5/2 The general solution of the homogeneous differential equation is
yh=c1+c2e5x/2 Find the particular solution of the non homogeneous differential equation
yp=Ax3+Bx2+Cx+De−4x
yp′=3Ax2+2Bx+C−4De−4x
yp′′=6Ax+2B+16De−4x Substitute
6Ax+2B+16De−4x
−15Ax2−10Bx−5C+20De−4x
=x2+5e−4x
−15A=1
6A−10B=0
2B−5C=0
36D=5
A=−1/15,B=−1/25,C=−2/125,D=5/36
yp=−151x3−251x2−1252x+365e−4x The general solution of the non homogeneous differential equation is
y=c1+c2e5x/2−151x3−251x2−1252x+365e−4x
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