In Problems 31-36, use the substitution X = e^t to transform the given Cauchy-Euler equation to a differential equation with constant coefficients. Solve the original equation by solving the new equation using the procedures in Sections 4.3-4.5. 31. y'' + 9y' + 20y 32. y' + 9xy' + 25y = 0 33. y' + [Oxy + 8y] x^2
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Given the Cauchy-Euler equation: $x^2y'' + 9xy' + 20y = 0$. Let $x = e^t$, then $t = \ln x$. We have: $$\frac{dx}{dt} = e^t \Rightarrow \frac{d^2x}{dt^2} = e^t$$ Now, we find the derivatives of $y$ with respect to $t$: $$y'(x) = \frac{dy}{dt} \cdot Show more…
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Use the substitution $x=e^{t}$ to transform the given Cauchy-Euler equation to a differential equation with constant coefficients. Solve the original equation by solving the new equation using the procedures in Sections 3.3-3.5. $$ x^{2} y^{\prime \prime}-9 x y^{\prime}+25 y=0 $$
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Use the substitution $x=e^{t}$ to transform the given Cauchy-Euler equation to a differential equation with constant coefficients. Solve the original equation by solving the new equation using the procedures in Sections 3.3-3.5. $$ x^{2} y^{\prime \prime}+9 x y^{\prime}-20 y=0 $$
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