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Okay, so I was sitting in my room wondering about the differential operator [tex]D[/tex]. Like for example, solving the equation

[tex]y\prime\prime+5y\prime+4y=0[/tex]

introduce [tex]D=\frac{d}{dx}[/tex]

[tex](D^2+5D+4)y=0[/tex]

so you can solve it by doing [tex]D=-4[/tex], [tex]D=-1[/tex]

[tex]e^-^4^x+e^-^x[/tex] is the solution

But how the hell do you solve? [tex]y\prime\prime+2xy=0[/tex]

or (with other variables involved?)

[tex]xy\prime+y=e^x^y[/tex]

or [tex]2yy\prime\prime=(y\prime)^2[/tex]

what is the method? What are the techniques?

[tex]y\prime\prime+5y\prime+4y=0[/tex]

introduce [tex]D=\frac{d}{dx}[/tex]

[tex](D^2+5D+4)y=0[/tex]

so you can solve it by doing [tex]D=-4[/tex], [tex]D=-1[/tex]

[tex]e^-^4^x+e^-^x[/tex] is the solution

But how the hell do you solve? [tex]y\prime\prime+2xy=0[/tex]

or (with other variables involved?)

[tex]xy\prime+y=e^x^y[/tex]

or [tex]2yy\prime\prime=(y\prime)^2[/tex]

what is the method? What are the techniques?

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