Question

Solve the following Cauchy Euler equations: 1. $x^2y'' - 2y = 0$ 2. $x^2y'' + xy' + 4y = 0$ 3. $3x^2y'' + 6xy' + y = 0$ 4. $2x^2y'' + 5xy' + y = x^2 - x$ 5. $x^2y'' + xy' - y = ln x$

          Solve the following Cauchy Euler equations:
1. $x^2y'' - 2y = 0$
2. $x^2y'' + xy' + 4y = 0$
3. $3x^2y'' + 6xy' + y = 0$
4. $2x^2y'' + 5xy' + y = x^2 - x$
5. $x^2y'' + xy' - y = ln x$

Added by Cheryl A.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Adi S

07/05/2023

Step 1

For the first equation, we have: $xy'' - 2y = 0$. We assume a solution of the form $y = x^r$, so $y' = rx^{r-1}$ and $y'' = r(r-1)x^{r-2}$. Plugging these into the equation, we get: $r(r-1)x^r - 2x^r = 0 \implies r^2 - r - 2 = 0$ Solving for $r$, we get $r = 2$ Show more…

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