Home › AMUEEE › Mathematics › Differential Equations › Solve $\dfrac{dy}{dx} = \dfrac{y + \sqrt{x^2 + y…
Solve $\dfrac{dy}{dx} = \dfrac{y + \sqrt{x^2 + y^2}}{x}$.
A$y + \sqrt{x^2 + y^2} = cx^2$
B$y^2 = cx$
C$\sqrt{x^2 + y^2} = c$
D$y = x + c$
Answer & Solution
Correct answer: A. $y + \sqrt{x^2 + y^2} = cx^2$
Homogeneous: put $y = vx$. After substitution: $x\,dv/dx = \sqrt{1+v^2}$ ⇒ $dv/\sqrt{1+v^2} = dx/x$. Integrating: $\log(v + \sqrt{1+v^2}) = \log(cx)$ ⇒ $v + \sqrt{1+v^2} = cx$. Back-substitute $v = y/x$: $y/x + \sqrt{1 + y^2/x^2} = cx$, which simplifies to $y + \sqrt{x^2+y^2} = cx^2$.
Related questions
Reducing $(2x-1)\dfrac{d^2y}{dx^2}-2\dfrac{dy}{dx}=0$ with $y=2$ and $\dfrac{dy}{dx}=3$ atA village grows at a rate proportional to its population. It was 20,000 in 1999 and 25,000For $\dfrac{d^2y}{dx^2}+3\dfrac{dy}{dx}+2y=6e^{x}+ in x$, the particular integral is:The complementary function of $\dfrac{d^2y}{dx^2}+3\dfrac{dy}{dx}+2y=6e^{x}+ in x$ is:Solutions of a differential equation form a set that is a:The power counted for the degree must be a whole number that is:An equation whose highest derivative is the third has order:The study at this stage is confined to equations that are: