Solve (D-2)x-(D+1)y=6e^3t

$$( D - 2 ) x - ( D + 1 ) y = 6 e ^ { 3 t }$$

Solve for D

$$\left\{\begin{matrix}D=\frac{2x+y+6e^{3t}}{x-y}\text{, }&x\neq y\\D\in \mathrm{R}\text{, }&x=-2e^{3t}\text{ and }y=-2e^{3t}\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply D-2 by x.

$$Dx-2x-\left(D+1\right)y=6e^{3t}$$

Use the distributive property to multiply D+1 by y.

$$Dx-2x-\left(Dy+y\right)=6e^{3t}$$

To find the opposite of Dy+y, find the opposite of each term.

$$Dx-2x-Dy-y=6e^{3t}$$

Add 2x to both sides.

$$Dx-Dy-y=6e^{3t}+2x$$

Add y to both sides.

$$Dx-Dy=6e^{3t}+2x+y$$

Combine all terms containing D.

$$\left(x-y\right)D=6e^{3t}+2x+y$$

The equation is in standard form.

$$\left(x-y\right)D=2x+y+6e^{3t}$$

Divide both sides by x-y.

$$\frac{\left(x-y\right)D}{x-y}=\frac{2x+y+6e^{3t}}{x-y}$$

Dividing by x-y undoes the multiplication by x-y.

$$D=\frac{2x+y+6e^{3t}}{x-y}$$

Solve for t

$$t=\frac{\ln(Dx-2x-Dy-y)-\ln(6)}{3}$$

$$\left(D>\frac{2x+y}{x-y}\text{ and }x>y\right)\text{ or }\left(x=y\text{ and }y<0\right)\text{ or }\left(D<\frac{2x+y}{x-y}\text{ and }x<y\right)$$