(Solved):     Consider the following differential equation \[ \left(x^{2}+2 x\right) y^{\prime ...

 

 

Consider the following differential equation \[ \left(x^{2}+2 x\right) y^{\prime \prime}-2(x+1) y^{\prime}+2 y=0 \] One solution of this equation \( y_{1}=x+1 \) is given. By the method of reduction of order, substitutions \( y=(x+1) v \) and \( z=v^{\prime} \) transform the given differential equation into a first order differential equation for \( \boldsymbol{z} \). What is the resulting first order differential equation? Your answer: \[ \begin{array}{l} x(x+2) z^{\prime}-2 z=0 \\ x(x+2) z^{\prime}+2 z=0 \\ x(x+1)(x+2) z^{\prime}-2 z=0 \\ x(x+1)(x+2) z^{\prime}+2 z=0 \end{array} \] None of the above ear answer