Solution of the differential equation x2 y d x=(x3+y3) dy with y(0)=1 is given by

Question

Solution of the differential equation x2 y d x=(x3+y3) dy with y(0)=1 is given by (A) x 2=2 y 2 ln y (B) x 3=3 y 3 ln y (C) y2=2 x ln x (D) y3=3 x3 ln x

Tardigrade Admin · Accepted Answer

Θ x2 y d x=(x3+y3) d y=x3 d y+y3 d y ⇒ x2(y d x-x d y)=y3 d y ⇒ (x2/y2) d((x/y))=(d y/y) ⇒ ( x 3/3 y 3)= ln y + C Θ x =0, y =1 ∴ 0=0+ c ⇒ c =0 ∴ x 3=3 y 3 ln y

Q. Solution of the differential equation $x^{2} y d x = (x^{3} + y^{3})$ dy with $y (0) = 1$ is given by

$x^{2} = 2 y^{2} ln y$

$x^{3} = 3 y^{3} ln y$

$y^{2} = 2 x ln x$

$y^{3} = 3 x^{3} ln x$

Q. Solution of the differential equation x2ydx=(x3+y3) dy with y(0)=1 is given by

x2=2y2lny

x3=3y3lny

y2=2xlnx

y3=3x3lnx

Θx2ydx=(x3+y3)dy=x3dy+y3dy ⇒x2(ydx−xdy)=y3dy ⇒y2x2​d(yx​)=ydy​ ⇒3y3x3​=lny+C Θx=0,y=1 ∴0=0+c⇒c=0 ∴x3=3y3lny

Q. Solution of the differential equation $x^{2} y d x = (x^{3} + y^{3})$ dy with $y (0) = 1$ is given by

$x^{2} = 2 y^{2} ln y$

$x^{3} = 3 y^{3} ln y$

$y^{2} = 2 x ln x$

$y^{3} = 3 x^{3} ln x$