Question

Solve the given system of differential equations.[Dx+Dy+(D+1)z=0Dx+y=e^{t}Dx+y-2z=50sin(2t)

First order differential equations

Solve the given system of differential equations.
$$Dx+Dy+(D+1)z=0$$

$$Dx+y=e^{t}$$

$$Dx+y-2z=50\sin(2t)$$

2021-02-22
The given system of differential equations is,
$$Dx+Dy+(D+1)z=0$$
$$Dx+y=e^{t}$$
$$Dx+y-2z=50\sin(2t)$$
Putting the value of $$Dx+y$$ in the third equation, we get,
$$Dx+y-2z=50\sin(2t)$$
$$\Rightarrow e^{t}-2z=50\sin(2t)$$
$$\Rightarrow z=\frac{1}{2}e^{t}-25\sin(2t)$$
Now differentiating both sides with respect to t, we get,
$$Dz=\frac{1}{2}e^{t}-50\cos(2t)$$
$$\Rightarrow (D+1)z=Dz+z=\bigg(\frac{1}{2}e^{t}-50\cos(2t)\bigg)+\frac{1}{2}e^{t}-25\sin(2t)$$
$$\Rightarrow (D+1)z=e^{t}-25\sin(2t)-50\cos(2t)$$
Putting $$Dx=e^{t}-y$$ and the value of $$(D+1)z$$ in first equation, we get,
$$e^{t}-y+Dy+e^{t}-25\sin 2t-50\cos 2t=0$$
$$\Rightarrow Dy-y=25\sin 2t + 50\cos 2t-2e^{t}$$
So the integrating factor of this differential equation is,
$$I.F=e^{\int(-1)dt}$$
$$=e^{-t}$$
Hence the solution is,
$$ye^{-t}=\int(25\sin2t+50\cos2t-2e^{t})e^{-t}dt$$
$$=\int25\sin2te^{-t}dt+\int50\cos2te^{-t}dt-2\int e^{t}e^{-t}dt$$
$$=-10e^{-t}\cos2t-5e^{-t}\sin2t+20e^{-t}\sin2t-10e^{-t}\cos2t-2t+C$$
$$\Rightarrow y=-10\cos2t-5\sin2t+20\sin2t-10\cos2t-2te^{t}+Ce^{t}$$
$$\Rightarrow y=-20\cos2t+15\sin2t-2te^{t}+Ce^{t}$$
Now putting this value of $$y$$ in the given second differential equation, we get: $$Dx=e^{t}-y$$
$$=e^{t}-(-20\cos2t+15\sin2t-2te^{t}+Ce^{t})$$
$$=e^{t}+20\cos2t-15\sin2t+2te^{t}-Ce^{t}$$
Integrating both sides, we get,
$$x=\int(e^{t}+20\cos2t-15\sin2t+2te^{t}-Ce^{t})dt$$
$$=e^{t}+10\sin2t+\frac{15}{2}\cos2t+2te^{t}-2e^{t}-Ce^{t}+D$$
$$=(-C-1)e^{t}+10\sin2t+\frac{15}{2}\cos2t+2te^{t}+D$$
$$=C'e^{t}+10\sin2t+\frac{15}{2}\cos2t+2te^{t}+D$$
Hence the required solution is,
$$x=C'e^{t}+10\sin2t+\frac{15}{2}\cos2t+2te^{t}+D$$
$$y=-20\cos2t+15\sin2t-2te^{t}+Ce^{t}$$
$$z=\frac{1}{2}e^{t}-25\sin2t$$
Note: For particular value $$C'=-\frac{1}{2} \text{ and } C=-\frac{1}{2}$$ these solutions satisfies
the given system of differential equations.
The required solution is,
$$x=C'e^{t}+10\sin2t+\frac{15}{2}\cos2t+2te^{t}+D$$
$$y=-20\cos2t+15\sin2t-2te^{t}+Ce^{t}$$
$$z=\frac{1}{2}e^{t}-25\sin2t$$