### Solution Found!

# Given that y c1 c2x2 is a two-parameter family of solutions of xy y 0 on the interval (

**Chapter 4, Problem 5**

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**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.

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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.