Name: Problem Set B UTSA ID (abc123): Date: Problem B.7 Consider the ODE y' + P(x)y = f(x). If P(x) and /(x) are both constants, then the ODE can be solved by separation of variables. Suppose P(x) = a and f (x) = b, where a and b are non-zero constants. We then have the ODE y' + ay = b. If y(0) = co, solve the initial-value problem. Verify by substitution that the explicit solution satisfies the ODE. y = (co -_ )e-ax +_ Verification by substitution leads to an identity, or 0 = 0. Problem B.& I*-order linear homogeneous ODEs having one dependent variable are of the form - + P(x)y = 0. Use the method of separation of variables to show that their general solution is of the form y = ce-J P(x)dx. Ly + P(x)y = 0 =-P(x)y JW= -[P(x)dx elntlyD+6 = e-/P(xjax y= Ge IP(x)dx