Chapter D1.4, Problem 33E

Special equations A special class of first-order linear equations have the form $a\left(t\right)y^{\prime }\left(t\right)+a^{\prime }\left(t\right)y\left(t\right)=f\left(t\right)$, 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

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

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

33.     $t{y}^{\prime }\left(t\right)+y=1+t,y\left(1\right)=4$