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A second-order Euler equation is one of the form ax2y00 C
Chapter , Problem 51(choose chapter or problem)
A second-order Euler equation is one of the form
\(a x^{2} y^{\prime \prime}+b x y^{\prime}+c y=0\)
where a, b, c are constants.
(a) Show that if x > 0, then the substitution v = ln x transforms Eq. (22) into the constant coefficient linear equation
\(a \frac{d^{2} y}{d v^{2}}+(b-a) \frac{d y}{d v}+c y=0 \quad (23)\)
with independent variable v.
(b) If the roots \(r_{1}\) and \(r_{2}\) of the characteristic equation of Eq. (23) are real and distinct, conclude that a general solution of the Euler equation in (22) is \(y(x)=c_{1} x^{r_{1}}+c_{2} x^{r_{2}}\).
Questions & Answers
QUESTION:
A second-order Euler equation is one of the form
\(a x^{2} y^{\prime \prime}+b x y^{\prime}+c y=0\)
where a, b, c are constants.
(a) Show that if x > 0, then the substitution v = ln x transforms Eq. (22) into the constant coefficient linear equation
\(a \frac{d^{2} y}{d v^{2}}+(b-a) \frac{d y}{d v}+c y=0 \quad (23)\)
with independent variable v.
(b) If the roots \(r_{1}\) and \(r_{2}\) of the characteristic equation of Eq. (23) are real and distinct, conclude that a general solution of the Euler equation in (22) is \(y(x)=c_{1} x^{r_{1}}+c_{2} x^{r_{2}}\).
ANSWER:Step 1 of 3
The given form of Euler equation
\(a x^{2} y^{\prime \prime}+b x y^{\prime}+c y=0\)
Now
\(\begin{aligned}
v & =\ln x \\
\frac{d v}{d x} & =\frac{1}{x}
\end{aligned}\)
Compute the \(y^{\prime} \& y^{\prime \prime}\) by using the chai rule.
\(y^{\prime}=\frac{d y}{d x}\)
Multiply and divide the above expression with dv
\(\begin{array}{l}
y^{\prime}=\frac{d y}{d v} \frac{d v}{d x} \\
y^{\prime}=\frac{1}{x} \frac{d y}{d v}
\end{array}\)