Solution Found!
Show that YI = x2 and Y2 = x3 are two different solutions
Chapter 2, Problem 2.1.29(choose chapter or problem)
Show that \(y_{1}=x^{2}\) and \(y_{2}=x^{3}\) are two different solutions of \(x^{2} y^{\prime \prime}-4 x y^{\prime}+6 y=0\), both satisfying the initial conditions \(y(0)=0=y^{\prime}(0)\). Explain why these facts do not contradic Theorem 2 (with respect to the guaranteed uniqueness).
Questions & Answers
QUESTION:
Show that \(y_{1}=x^{2}\) and \(y_{2}=x^{3}\) are two different solutions of \(x^{2} y^{\prime \prime}-4 x y^{\prime}+6 y=0\), both satisfying the initial conditions \(y(0)=0=y^{\prime}(0)\). Explain why these facts do not contradic Theorem 2 (with respect to the guaranteed uniqueness).
ANSWER:Step 1 of 3
Given that differential equation as \(x^{2} y^{\prime \prime}-4 x y^{\prime}+6 y=0\)
\(\begin{array}{l}
y_{1}=x^{2} \\
y_{1}^{\prime}(x)=2 x \\
y_{1}^{\prime \prime}=2
\end{array}\)
\(y_{1}\) satisfies the given differential equation,
\(x^{2} y_{1} "-4 x y_{1}^{\prime}+6 y_{1}=2 x^{2}-8 x^{2}+6 x^{2}\)
Hence \(y_{1}\) is the solution of the differential equation