Euler Equations — Question 1

Question 1 – Solution

Strategy. Logarithmic coordinates turn powers into exponentials and make the characteristic equation systematic.

Step 1: Transform derivatives correctly. The chain rule gives $Y'=xy'$ and $Y''=xy'+x^2y''$, hence $$\boxed{xy'=Y',\qquad x^2y''=Y''-Y'.}$$ The equation becomes $Y''-3Y'-4Y=0$, with characteristic polynomial $m^2-3m-4=(m-4)(m+1)$.

Step 2: Solve and verify the data. The powers $x^4,x^{-1}$ form a pair with $W=x^4(-x^{-2})-4x^3x^{-1}=-5x^2\ne 0$ for $x>0$. Writing $y=C_1x^4+C_2x^{-1}$ gives $C_1+C_2=2$, $4C_1-C_2=3$, so $$\boxed{y=x^4+x^{-1}.}$$ At $1$, its value is $2$ and derivative $4x^3-x^{-2}$ is $3$.

Step 3: Identify the maximal interval. The normalized coefficients are continuous on $(0,\infty)$, and the explicit solution exists throughout it. Since $y\to+\infty$ as $x\to 0^+$, no finite continuation through $0$ is possible. Thus the maximal real interval through $1$ is $\boxed{(0,\infty)}$. At infinity, $y\sim x^4\to+\infty$; at zero, $y\sim x^{-1}$.

Step 4: Find the unique minimum. The equation $y'=0$ is $4x^5=1$, giving $x_*=4^{-1/5}$. Since $y''=12x^2+2x^{-3}>0$, it is the unique global minimum. Using $x_*^{-1}=4x_*^4$, $$\boxed{y(x_*)=5\cdot 4^{-4/5}.}$$ One mode increases and the other decreases. Their derivatives cancel at exactly one point, although neither mode has a stationary point of its own.