The solution of frac{dy}{dx} = frac{ax + b}{c | vedclass

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(C) Given the differential equation: $\frac{dy}{dx} = \frac{ax + b}{cy + d}$.By separating the variables,we get: $(cy + d) dy = (ax + b) dx$.Integrati

Vedclass · Accepted Answer

$a = 0, c 
eq 0$ or $c = 0, a 
eq 0$

Vedclass · Answer

(C) Given the differential equation: $\frac{dy}{dx} = \frac{ax + b}{cy + d}$.By separating the variables,we get: $(cy + d) dy = (ax + b) dx$.Integrating both sides: $\int (cy + d) dy = \int (ax + b) dx$.This yields: $\frac{c y^2}{2} + dy = \frac{a x^2}{2} + bx + K$,where $K$ is the constant of integration.Rearranging the terms: $c y^2 - a x^2 + 2dy - 2bx - 2K = 0$.For this equation to represent a parabola,one of the squared terms ($y^2$ or $x^2$) must be present,but not both with the same sign and magnitude (which would represent a circle or ellipse).Specifically,if $c = 0$ and $a 
eq 0$,the equation becomes $-a x^2 + 2dy - 2bx - 2K = 0$,which is a parabola opening along the $x$-axis.If $a = 0$ and $c 
eq 0$,the equation becomes $c y^2 + 2dy - 2bx - 2K = 0$,which is a parabola opening along the $y$-axis.Thus,the condition is $a = 0, c 
eq 0$ or $c = 0, a 
eq 0$.

The solution of $\frac{dy}{dx} = \frac{ax + b}{cy + d}$ represents a parabola if

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