Differential Equations Question 99
Question: To reduce the differential equation $ \frac{dy}{dx}+P(x)y=Q(x).y^{n} $ to the linear form, the substitution is
[UPSEAT 2004]
Options:
A) $ v=\frac{1}{y^{n}} $
B) $ v=\frac{1}{{y^{n-1}}} $
C) $ v=y^{n} $
D) $ v={y^{n-1}} $
Show Answer
Answer:
Correct Answer: B
Solution:
An equation of the form $\frac{dy}{dx}+Py=Qu^n$ where P and Qare functions ofx alone or constants, is called Bernoulli’s equation.
Divide both the sides by yn, we get
$y^{−n}\frac{dy}{dx}+Py^{−n+1}=Q $
Put
$y^{−n+1}=z⇒(−n+1)y^{−n}\frac{dy}{dx}=\frac{dz}{dx}$.
The equation reduces to $\frac{1}{1−n}\frac{dz}{dx}+Pz=Q⇒\frac{dz}{dx}+(1−n)Pz=Q$
Which is linear in z and can be solved in the usual manner.So the substitution is z=v=$\frac{1}{y^{n−1}}$
BETA
