The solution of the differential equation (d y/d x)=(y2 + x ln x/2 x y) is (where, c is the constant of integration)

Question

The solution of the differential equation (d y/d x)=(y2 + x ln x/2 x y) is (where, c is the constant of integration) (A) 2x2=y(ln x)2+2cy (B) 2y2=x(ln x)2+2cx (C) x2=y(ln x)2+c (D) 2y2=(x/y)(ln x)2+cx

Tardigrade Admin · Accepted Answer

2xy(d y/d x)=y2+xln x 2y(d y/d x)-(y2/x)=ln x Put, y2=t⇒ 2y(d y/d x)=(d t/d x) (d t/d x)-(t/x)=ln x I.F. =e∫ (- 1/x) d x=e- l n x=(1/x) The solution is (t/x)=∫ (ln x/x) d x (t/x)=((ln x)2/2)+c (y2/x)=((ln x)2/2)+c 2y2=x(ln x)2+2cx

Q. The solution of the differential equation $\frac{d y}{d x} = \frac{y ^{2} + x l n x}{2 x y}$ is (where, $c$ is the constant of integration)

$2 x^{2} = y (l n x)^{2} + 2 cy$

$2 y^{2} = x (l n x)^{2} + 2 c x$

$x^{2} = y (l n x)^{2} + c$

$2 y^{2} = \frac{x}{y} (l n x)^{2} + c x$

Q. The solution of the differential equation dxdy​=2xyy2+xlnx​ is (where, c is the constant of integration)

2x2=y(lnx)2+2cy

2y2=x(lnx)2+2cx

x2=y(lnx)2+c

2y2=yx​(lnx)2+cx

2xydxdy​=y2+xlnx 2ydxdy​−xy2​=lnx Put, y2=t⇒2ydxdy​=dxdt​ dxdt​−xt​=lnx I.F. =e∫x−1​dx=e−lnx=x1​ The solution is xt​=∫xlnx​dx xt​=2(lnx)2​+c xy2​=2(lnx)2​+c 2y2=x(lnx)2+2cx

Q. The solution of the differential equation $\frac{d y}{d x} = \frac{y ^{2} + x l n x}{2 x y}$ is (where, $c$ is the constant of integration)

$2 x^{2} = y (l n x)^{2} + 2 cy$

$2 y^{2} = x (l n x)^{2} + 2 c x$

$x^{2} = y (l n x)^{2} + c$

$2 y^{2} = \frac{x}{y} (l n x)^{2} + c x$