Chapter 6 - Differential Equations - 6.4 Exercises - Page 430: 49

$$\eqalign{ & y = - 2\cot x + \left( {\sin \left( 1 \right) + 2\cos \left( 1 \right)} \right)\csc x{\text{ }}\left( {{\text{Particular solution 1}}} \right) \cr & y = - 2\cot x + \left( {2\cos \left( 3 \right) - \sin \left( 3 \right)} \right)\csc x{\text{ }}\left( {{\text{Particular solution 2}}} \right) \cr} $$

$$\eqalign{ & \frac{{dy}}{{dx}} + \left( {\cot x} \right)y = 2 \cr & {\text{The differential equation has the form }}\frac{{dy}}{{dx}} + P\left( x \right)y = Q\left( x \right) \cr & {\text{With }}P\left( x \right) = \cot x{\text{ and }}Q\left( x \right) = 2 \cr & {\text{Find the integrating factor }}I\left( x \right) = {e^{\int {P\left( x \right)} dx}} \cr & I\left( x \right) = {e^{\int {\cot xdx} }} = {e^{\ln \left| {\sin x} \right|}} = \sin x \cr & {\text{Multiply the differential equation by the integrating factor}} \cr & \sin x\left[ {\frac{{dy}}{{dx}} + \left( {\cot x} \right)y} \right] = \sin x\left( 2 \right) \cr & {\text{Write the left side in the form }}\frac{d}{{dx}}\left[ {I\left( x \right)y} \right] \cr & \frac{d}{{dx}}\left[ {y\sin x} \right] = 2\sin x \cr & {\text{Integrate both sides}} \cr & y\sin x = - 2\cos x + C \cr & {\text{Solve for }}y \cr & y = - \frac{{2\cos x}}{{\sin x}} + C\csc x \cr & y = - 2\cot x + C\csc x \cr & {\text{Use the initial condition }}\left( {1,1} \right) \cr & 1 = - 2\cot \left( 1 \right) + C\csc \left( 1 \right) \cr & 1 = - 2\cot \left( 1 \right) + C\csc \left( 1 \right) \cr & C = \sin \left( 1 \right) + 2\cos \left( 1 \right) \cr & y = - 2\cot x + \left( {\sin \left( 1 \right) + 2\cos \left( 1 \right)} \right)\csc x{\text{ }}\left( {{\text{Particular solution 1}}} \right) \cr & {\text{Use the initial condition }}\left( {3, - 1} \right) \cr & - 1 = - 2\cot \left( { - 1} \right) + C\csc \left( { - 1} \right) \cr & - 1 = 2\cot \left( 1 \right) - C\csc \left( 1 \right) \cr & C = 2\cos \left( 3 \right) - \sin \left( 3 \right) \cr & y = - 2\cot x + \left( {2\cos \left( 3 \right) - \sin \left( 3 \right)} \right)\csc x{\text{ }}\left( {{\text{Particular solution 2}}} \right) \cr & \cr & {\text{Graph}} \cr} $$