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Given that y c1 c2x2 is a two-parameter family of solutions of xy y 0 on the interval (
Chapter 4, Problem 5(choose chapter or problem)
Given that \(y=c_{1}+c_{2} x^{2}\) is a two-parameter family of solutions of \(x y^{\prime \prime}-y^{\prime}=0\) on the interval \((-\infty, \quad)\), show that constants \(c_1\) and \(c_2\) cannot be found so that a member of the family satisfies the initial conditions \(y(0)=0, y^{\prime}(0)=1\). Explain why this does not violate Theorem 4.1.1.
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Questions & Answers
QUESTION:
Given that \(y=c_{1}+c_{2} x^{2}\) is a two-parameter family of solutions of \(x y^{\prime \prime}-y^{\prime}=0\) on the interval \((-\infty, \quad)\), show that constants \(c_1\) and \(c_2\) cannot be found so that a member of the family satisfies the initial conditions \(y(0)=0, y^{\prime}(0)=1\). Explain why this does not violate Theorem 4.1.1.
ANSWER:Step 1 of 3
Consider the given equation.
\(y=c_{1}+c_{2} x^{2}\)
The initial condition \(y(0)\) gives,
\(\begin{aligned}
c_{1}+c_{2}(0)^{2} & =0 \\
c_{1} & =0
\end{aligned}\)
The initial condition \(y^{\prime}(0)=1\) gives,
\(\begin{aligned}
y^{\prime}(x) & =2 c_{2} x \\
y^{\prime}(0) & =2 c_{2}(0) \\
1 & =0
\end{aligned}\)
The above obtained relation is not true which means the constants \(c_{1}\) and \(c_{2}\) cannot be found so that a member of family satisfies the initial conditions.