Find the general solution of the Euler equation x^2y'' + xy'…

Find the general solution of the Euler equation $x^2y'' + xy' + y = 1$.

Step 1: The Euler substitution $x=e^t$

An Euler (equidimensional) equation is one in which the coefficient of $y^{(k)}$ is proportional to $x^k$; such equations are always reduced to constant-coefficient linear equations by the substitution $x=e^t$, i.e.\ $t=\ln x$ (for $x>0$). By the chain rule,

\[ \frac{dy}{dx} = \frac{dy}{dt}\cdot\frac{dt}{dx} = \dot y\cdot\frac1x \implies x\,y' = \dot y, \] where the dot denotes $d/dt$. Differentiating $xy'=\dot y$ again with respect to $x$ (using $\dfrac{d}{dx}=\dfrac1x\dfrac{d}{dt}$ on the right side and the product rule on the left) gives, after simplification, the standard second identity

\[ x^2y'' = \ddot y - \dot y. \]

Step 2: Transformed equation

Substituting $x^2y''=\ddot y-\dot y$ and $xy'=\dot y$ into $x^2y''+xy'+y=1$: $(\ddot y - \dot y) + \dot y + y = 1 \implies \ddot y + y = 1.$

Step 3: Solving the constant-coefficient equation in $t$

Characteristic equation: $r^2+1=0\implies r=\pm i$, so

\[ Y_h = C_1\cos t + C_2\sin t. \]

For the particular solution, the right-hand side is the constant $1$, and $r=0$ is not a root, so try $y^{*}=A$: substituting gives $A=1$. Hence

\[ y = C_1\cos t + C_2\sin t + 1. \]

Step 4: Return to the variable $x$

Since $t=\ln x$, $\cos t=\cos(\ln x)$ and $\sin t=\sin(\ln x)$, giving the general solution $y = C_1\cos(\ln x) + C_2\sin(\ln x) + 1$.

Step 5: Verification

Take, for concreteness, $y=\cos(\ln x)+1$ (the $\sin(\ln x)$ term checks identically by the same computation). By the chain rule, $y' = -\frac{\sin(\ln x)}{x}, \qquad y'' = \frac{-\cos(\ln x)+\sin(\ln x)}{x^2}.$

Then \[x^2y'' + xy' + y = \bigl[-\cos(\ln x)+\sin(\ln x)\bigr] + \bigl[-\sin(\ln x)\bigr] + \bigl[\cos(\ln x)+1\bigr] = 1.\ \checkmark\]

The $\cos(\ln x)$ and $\sin(\ln x)$ contributions cancel exactly, leaving only the particular solution's contribution of $1$; by linearity, the full general solution satisfies the original equation for all $C_1,C_2$.