Mathematics: Use the substitution of x=e^{t} to transform the given cauchy-euler differential equation to a differential equation with constant coefficients then solve that by any method. (finding the characterristic (auxiliary) equation or variation of parameters). do not forget to re-subtitute for x after you solve the equation. x^{2}y^{ }+9xy^{ }-20y=0

both are negativeStep-by-step explanation:...

Use the substitution of x=e^{t} to transform the given

Ответ:

ashneedshelp1251

13.03.2022

By the chain rule,

$\dfrac{\mathrm dy}{\mathrm dx}=\dfrac{\mathrm dy}{\mathrm dt}\dfrac{\mathrm dt}{\mathrm dx}\implies\dfrac{\mathrm dy}{\mathrm dt}=x\dfrac{\mathrm dy}{\mathrm dx}$

which follows from $x=e^t\implies t=\ln x\implies\dfrac{\mathrm dt}{\mathrm dx}=\dfrac1x$ .

$\dfrac{\mathrm dy}{\mathrm dt}$ is then a function of ; denote this function by f(x) . Then by the product rule,

$\dfrac{\mathrm d^2y}{\mathrm dx^2}=\dfrac{\mathrm d}{\mathrm dx}\left[\dfrac1x\dfrac{\mathrm dy}{\mathrm dt}\right]=-\dfrac1{x^2}\dfrac{\mathrm dy}{\mathrm dt}+\dfrac1x\dfrac{\mathrm df}{\mathrm dx}$