Question

Find the general solution of the differential equation y^(5) −2y^(4) + y^(3) = 0.

Answer 1

$y^{(5)}-2y^{(4)}+y^{(3)}=0$

We can reduce the order of the ODE by substituting $v(x)=y^{(3)}(x)$, so that $v'(x)=y^{(4)}(x)$ and $v''(x)=y^{(5)}(x)$. Then

$v''-2v'+v=0$

has characteristic equation

$r^2-2r+1=(r-1)^2=0$

with root $r=1$, which has multiplicity 2, so that the characteristic solution is

$v_c=C_1e^x+C_2xe^x$

Integrate both sides to solve for $y''(x)$:

$y''=C_1e^x+C_2e^x(x-1)+C_3$

$y''=C_1e^x+C_2xe^x+C_3$

Integrate again to solve for $y'(x)$:

$y'=C_1e^x+C_2e^x(x-1)+C_3x+C_4$

$y'=C_1e^x+C_2xe^x+C_3x+C_4$

And one last time to solve for $y(x)$:

$y=C_1e^x+C_2e^x(x-1)+\dfrac{C_3}2x^2+C_4x+C_5$

$\boxed{y(x)=C_1e^x+C_2e^x+C_3x^2+C_4x+C_5}$