If y=(x/ ln |c x|) (where c is an arbitrary constant) is the general solution of the differential equation (d y/d x)=(y/x) +φ((x/y)) then the function φ((x/y)) is :

Question

If y=(x/ ln |c x|) (where c is an arbitrary constant) is the general solution of the differential equation (d y/d x)=(y/x) +φ((x/y)) then the function φ((x/y)) is : (A) (x2/y2) (B) -(x2/y2) (C) (y2/x2) (D) -(y2/x2)

Tardigrade Admin · Accepted Answer

ln c + ln | x |=( x / y ) text diff. w.r.t. x , (1/ x )=( y - xy 1/ y 2) (y2/x)=y-x (d y/d x) (d y/d x)=(y/x)-(y2/x2) ⇒ φ((x/y))=-(y2/x2) ⇒ D

Q. If $y = \frac{x}{l n ∣ c x ∣}$ (where $c$ is an arbitrary constant) is the general solution of the differential equation $\frac{d y}{d x} = \frac{y}{x}$ $+ ϕ (\frac{x}{y})$ then the function $ϕ (\frac{x}{y})$ is :

$\frac{x ^{2}}{y ^{2}}$

$- \frac{x ^{2}}{y ^{2}}$

$\frac{y ^{2}}{x ^{2}}$

$- \frac{y ^{2}}{x ^{2}}$

$ln c + ln ∣ x ∣ = \frac{x}{y}$
$diff. w.r.t. x, \frac{1}{x} = \frac{y - x y _{1}}{y ^{2}}$
$\frac{y ^{2}}{x} = y - x \frac{d y}{d x}$
$\frac{d y}{d x} = \frac{y}{x} - \frac{y ^{2}}{x ^{2}} \Rightarrow ϕ (\frac{x}{y}) = - \frac{y ^{2}}{x ^{2}} \Rightarrow D$

Q. If y=ln∣cx∣x​ (where c is an arbitrary constant) is the general solution of the differential equation dxdy​=xy​ +ϕ(yx​) then the function ϕ(yx​) is :

y2x2​

−y2x2​

x2y2​

−x2y2​