Chapter D1.3, Problem 41E

Implicit solutions for separable equations For the following separable equations, carry out the indicated analysis.

1. a. Find the general solution of the equation.
2. b. Find the value of the arbitrary constant associated with each initial condition. (Each initial condition requires a different constant.)
3. c. Use the graph of the general solution that is provided to sketch the solution curve for each initial condition.

41.      $e^{-y/2}{y}^{\prime }\left(x\right)=4x\sin x^2-x;y\left(0\right)=0,y\left(0\right)=\ln \left(\frac{1}{4}\right),y\left(\sqrt{\frac{\text{π}}{2}}\right)=0$