Question

Solve dy/dx-4y=2x^2 using the method exploit the superstition.

Answer 1

$$y=uv$$

$$y'=u'v+uv'$$

Substitute

$$u'v+uv'-4uv=2x^2$$

$$uv'+v(u'-4u)=2x^2$$

Put the $v$  term equal to zero

$$u'-4u=0$$

$$\dfrac{du}{u}=4dx$$

Integrate

$$\int \dfrac{du}{u}=\int4dx$$

$$u=Ce^{4x}$$

Then

$$Ce^{4x}v'=2x^2$$

$$dv=\dfrac{2}{C}x^2 e^{-4x}dx$$

Integrate

$$\int dv=\int\dfrac{2}{C}x^2 e^{-4x}dx$$

$$v=-\dfrac{1}{2C}x^2e^{-4x}-\dfrac{1}{4C}xe^{-4x}-\dfrac{1}{16C}e^{-4x}+C_1$$

$$y=-\dfrac{1}{2}x^2-\dfrac{1}{4}x-\dfrac{1}{16}+C_2e^{4x}$$