22 Jun 2025

Cauchy-Euler and Simultaneous Differential Equations

Power-law solutions of the Cauchy-Euler equation and normal combinations for coupled first- and second-order systems.

bsc semester-ii mathematical-physics cauchy-euler-equation simultaneous-equations

Cauchy-Euler equation

A second-order Cauchy-Euler equation has the form

\[ax^2y''+bxy'+cy=0, \qquad x>0.\]

Every derivative of a power reduces its exponent by one, while the prefactor restores it. With $y=x^m$,

\[y'=mx^{m-1}, \qquad y''=m(m-1)x^{m-2}.\]

Substitution gives

\[x^m\left[am(m-1)+bm+c\right]=0.\]

Thus the indicial equation is

\[\boxed{am(m-1)+bm+c=0}.\]

For

\[x^2y''-3xy'+4y=0,\]

the indicial polynomial is

\[m(m-1)-3m+4=(m-2)^2.\]

The repeated-root solution is

\[\boxed{y=x^2(C_1+C_2\ln x)}.\]

The logarithm follows directly from the change of variable $t=\ln x$. Since

\[\frac{dy}{dx}=\frac1x\frac{dy}{dt},\]

another differentiation gives

\[\frac{d^2y}{dx^2} =\frac1{x^2}\left(\frac{d^2y}{dt^2}-\frac{dy}{dt}\right).\]

The original equation becomes

\[\frac{d^2y}{dt^2}-4\frac{dy}{dt}+4y=0,\]

whose repeated root is $2$. Hence $y=(C_1+C_2t)e^{2t}=x^2(C_1+C_2\ln x)$.

Simultaneous first-order equations

Consider

\[x'=3x+y, \qquad y'=x+3y.\]

Add and subtract the equations. With

\[u=x+y, \qquad v=x-y,\]

we obtain

\[u'=4u, \qquad v'=2v.\]

Therefore $u=Ae^{4t}$ and $v=Be^{2t}$. Solving $u=x+y$, $v=x-y$ for the original variables gives

\[\boxed{ x=\frac12(Ae^{4t}+Be^{2t}), \qquad y=\frac12(Ae^{4t}-Be^{2t})}.\]

Simultaneous second-order equations

For

\[x''+2x-y=0, \qquad y''+2y-x=0,\]

the same combinations decouple the system. Addition gives

\[u''+u=0,\]

while subtraction gives

\[v''+3v=0.\]

Thus

\[u=A\cos t+B\sin t, \qquad v=C\cos(\sqrt3t)+D\sin(\sqrt3t),\]

and

\[\boxed{x=\frac{u+v}{2},\qquad y=\frac{u-v}{2}}.\]

The combinations \(u=x+y\) and \(v=x-y\) isolate the in-phase and out-of-phase motions. Their equations contain no remaining coupling.

Direct substitution into both coupled systems is carried out in the Unit II Maxima worksheet.