ISI2015-MMA-88

Question

Let $f(x)$ be a given differentiable function. Consider the following differential equation in $y$ $$f(x) \frac{dy}{dx} = yf’(x)-y^2.$$ The general solution of this equation is given by

• A. $y=-\frac{x+c}{f(x)}$
• B. $y^2=\frac{f(x)}{x+c}$
• C. $y=\frac{f(x)}{x+c}$
• D. $y=\frac{\left[f(x)\right]^2}{x+c}$

Comments

Try option C

Answer 1

C)

Follow the following steps,

1. Divide throughout by $f(x)$ and $y^2$.
2. Consider $z=\frac{1}{y}$, you will get standard differential equation in $z$.
3. Remember $$\int \frac{f’(x)}{f(x)}=\ln|f(x)| +c$$ you can ingore mod.
4. Solve for $z$ and then sustitute $z=\frac{1}{y}$

Comments

Or we may consider z=f(x)/y in step 2