Separation Of Variables And Integrating Factors Exercise

CLAREMONT MCKENNA COLLEGE Separation of Variables and Integrating Factors Solve for �y=(12�2)12 Solution: Derive y with respect to x, �(����)=(12�2)12 Cross-multiply and integrate both sides, (��(12�2)12)=(1�)�� Integrating both terms, beginning with (�(12�2)12), let =(2�2)12=2�,�=2����=2 Then, substituting in (�(12�2)12),(2(12�2)12)=(��) Solving for (2(12�2)12)=(�), 12(�(12)12)=(���), Because of the property of reverse trig functions for sine, d/dx becomes: ���(sin1())=�/(12)^1/2 We can simplify it and solve for the integral, 12sin1()=ln|�|+�12sin1(2�)=ln|�|+ln|�| 12sin1(2�)=ln|��| Cross-multiply and simplify, sin1(2�)=2ln|��| 2�=sin(2ln|��|) �= sin(2ln|��|)2