Chapter 2 Solvable Equations

Recall Sec 2.1 – 1 st Order Linear Equations  First solvable class of equations  The equation must be able to be expressed in the form  The solution “trick” depends on the chain rule and product rule  We write the integrating factor  Multiply the entire original equation by  And so the original equation becomes (rewritten)  This doesn’t look like much, but…

Left-hand side  Compute using product & chain rules  Prod rule: =  So LHS of the rewritten eq is  And when we integrate it, we get

Returning to the original equation rewritten,  If we integrate both sides wrt x,  Which gives us  Or,  This is the general form of the solution to the first order linear DE

Example  Solve  Divide by x^2+1 to get the standard form  Identify p(x), find rho  Use the formula to solve for y