Let y=y(x) satisfy the differential equation (2 x y+x^2 y+ y^3 3 ) d x+ (x^2+y^2 ) d y=0 . If y(1)=1 and the value of (y(0))^3=k e(k N) . Find k .

Step-by-step solution

aligned & (2 x y d x+x^2 d y )+x^2 y d x+ ( y^3 3 d x+y^2 d y )=0 & x^2 y=t and y^3 3 =u &(d t+t d x)+(u d x+d u)=0 & d t+d u=-(t+u) d x & d t+d u t+u =- d x & (t+u)=-x+C x^2 y+ y^3 3 =k^ e^ -x at x=1, y=1 & 4 3 = k^ e k^ = 4 e 3 aligned aligned x^2 y+ y^3 3 & = 4 e 3 ; put x=0 y^3 3 & = 4 e 3 y^3(0) & =4 e k=4 aligned