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Question Number 183210 by Mastermind last updated on 23/Dec/22

$$\frac{{\mathrm{d}}^3\mathrm{y}}{{\mathrm{dx}}^3}+4\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2}+\frac{\mathrm{dy}}{\mathrm{dx}}-6\mathrm{y}=0$$$$$$$$$$$$\mathrm{M}.\mathrm{m}$$

Answered by aleks041103 last updated on 23/Dec/22

$$y=e^{rx}\Rightarrow (r^3+4r^2+r-6)y=0$$$$\Rightarrow r^3+4r^2+r-6=0$$$$r^3-r^2+5r^2-5r+6r-6=0$$$$r^2(r-1)+5r(r-1)+6(r-1)=0$$$$\Rightarrow (r-1)(r^2+5r+6)=0$$$$(r-1)(r^2+2r+3r+6)=0$$$$(r-1)(r(r+2)+3(r+2))=0$$$$(r-1)(r+2)(r+3)=0$$$$\Rightarrow r=-3,-2,1$$$$\Rightarrow y\left(x\right)={Ae}^{-3x}+{Be}^{-2x}+{Ce}^x$$

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