Chapter D1.4, Problem 48E

General first-order linear equations Consider the general first-order linear equation y′(t) + a(t)y(t) = f(t). This equation can be solved, in principle, by defining the integrating factor p(t) = exp(òa(t) dt). Here is how the integrating factor works. Multiply both sides of the equation by p (which is always positive) and show that the left side becomes an exact derivative. Therefore, the equation becomes

     $p(t)(y^{\prime }(t)+a(t)y(t))=\frac{d}{dt}(p(t)y(t))=p(t)f(t).$

Now integrate both sides of the equation with respect to t to obtain the solution. Use this method to solve the following initial value problems. Begin by computing the required integrating factor.

48.    y′(t) + 2ty(t) = 3t, y(0) = 1