Chapter D1.4, Problem 35E

Special equations A special class of first-order linear equations have the form a(t)y′(t) + a′(t)y(t) = f(t), where a and f are given functions of t. Notice that the left side of this equation can be written as the derivative of a product, so the equation has the form

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

Therefore, the equation can be solved by integrating both sides with respect to t. Use this idea to solve the following initial value problems.

35.     $e^{-t}{y}^{\prime }(t)-e^{-t}y=e^{2t},y(0)=4$