In this problem we consider an equation in differential form mdx+ndy=0. the equation (4y+(5x^4)e^(? 4x))dx+(1? 4y^3(e^(? =0 in differential form m˜dx+n˜dy=0 is not exact. indeed, we have m˜y? n˜x= for this exercise we can find an integrating factor which is a function of x alone since m˜y? n˜xn˜= can be considered as a function of x alone. namely we have ? (x)= multiplying the original equation by the integrating factor we obtain a new equation mdx+ndy=0 where m= n= which is exact since my= nx= are equal. this problem is exact. therefore an implicit general solution can be written in the form f(x, y)=c where f(x, y)= finally find the value of the constant c so that the initial condition y(0)=1. c= .