# 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.

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.

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.