A second-order Euler equation is one of the form
ax²y''+bxy'+cy=0
where a, b, and c are constants. a) Show that if x>0 then the substitution v=lnx transforms the above equation into the constant-coefficient linear equation
a(d²y/dv²) + (b - a)(dy/dv) + cy = 0
with the independent variable v.
b) if the roots r1 and r2 of the characteristic equation above are real and distinct, conclude that a general solution of the Euler equation above is y(x) = (c1)x^(r1) + (c2)x^(r2).
ax²y''+bxy'+cy=0
where a, b, and c are constants. a) Show that if x>0 then the substitution v=lnx transforms the above equation into the constant-coefficient linear equation
a(d²y/dv²) + (b - a)(dy/dv) + cy = 0
with the independent variable v.
b) if the roots r1 and r2 of the characteristic equation above are real and distinct, conclude that a general solution of the Euler equation above is y(x) = (c1)x^(r1) + (c2)x^(r2).
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just calculations...dy / dx = [ dy / dv ] [ dv / dx ] = [1/x] dy / dv
and d²y / dx² = d[ dy/dx ] / dx = d [ (1/x) dy/dv ]/ dv * [ dv /dx ] = ( 1 / x² ) d²y / dv²
you certainly , being in DE , can finish the problem
and d²y / dx² = d[ dy/dx ] / dx = d [ (1/x) dy/dv ]/ dv * [ dv /dx ] = ( 1 / x² ) d²y / dv²
you certainly , being in DE , can finish the problem
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