Question #285084

Solve the partial differential equation (3D² +10DD'+ 3D ^1^2)z^ =e^x-y

1
Expert's answer
2022-01-06T11:28:45-0500

Since, the given equation is unclear, we assume it as:

Solve(D2+4D+3)y=e3x\quad Solve \left(D^{2}+4 D+3\right) y=e^{-3 x}

Solution: Given equation is

(D2+4D+3)y=e3x\left(D^{2}+4 D+3\right) y=e^{-3 x}

A E is D2+4D+3=0\quad D^{2}+4 D+3=0

(D+1)(D+3)=0D=1,D=3CF=c1ex+c2e3xPI=1(D+1)(D+3)e3x\begin{aligned} &(D+1)(D+3)=0 \quad D=-1, \quad D=-3 \\ \therefore \quad & C F=c_{1} e^{-x}+c_{2} e^{-3 x} \\ & P I=\frac{1}{(D+1)(D+3)} e^{-3 x} \end{aligned}

put D=-3 in f(D) then

PI=10e3x method fails PI=e3x(3+1)(D3+3){1}=e3x(2)1D{1}=e3x2xG.S.=CF+PI\begin{aligned} P I &=\frac{1}{0} e^{-3 x} \quad \therefore \text { method fails } \\ \therefore \quad P I &=\frac{e^{-3 x}}{(-3+1)(D-3+3)}\{1\}=\frac{e^{-3 x}}{(-2)} \frac{1}{D}\{1\}=\frac{e^{-3 x}}{-2} x \\ G . S . &=C F+P I \end{aligned}

y=c1ex+c2e3xe3x2xy=c_{1} e^{-x}+c_{2} e^{-3 x}-\frac{e^{-3 x}}{2} x



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