Question

Can someone assist me to solve how to use power series to solve the differential equation dy/dx = y - x - 1.

Answer 1

separately*

Answer 2

\[\frac{dy(x)}{dx}-y(x)\text{ = }-x-1\]
\[\text{Let }\mu (x)\text{ = }\exp (\int\limits -1 \, dx)\text{ = }e^{-x}\]
\[\text{Multiply both sides by \mu(x)}\]
\[e^{-x} \frac{dy(x)}{dx}-e^{-x} y(x)\text{ = }-e^{-x} (x+1)\]
Sub \[-e^{-x}\text{ = }\frac{d\text{}}{dx}\left(e^{-x}\right)\]
\[e^{-x} \frac{dy(x)}{dx}+\frac{d\text{}}{dx}\left(e^{-x}\right) y(x)\text{ = }-e^{-x} (x+1)\]
Use reverse product rule
\[\frac{d\text{}}{dx}\left(e^{-x} y(x)\right)\text{ = }-e^{-x} (x+1)\]
\[\int\limits \frac{d\text{}}{dx}\left(e^{-x} y(x)\right) \, dx\text{ = }\int\limits -e^{-x} (x+1) \, dx\]
\[e^{-x} y(x)\text{ = }-e^{-x} (-x-2)+c_1\]
\[y(x)\text{ = }x+c_1 e^x+2\]

Answer 3

Thanks that really helped