AP EAMCET · Maths · Differential Equations
If a and b are arbitrary constants, then the differential equation corresponding to the family of curves \(y=\tan (a x+b)\) is
- A \(\left(1+x^2\right) y_2-2 y y_1+y=0\)
- B \(\left(1+y^2\right) y_2-2 y y_1^2=0\)
- C \(\left(1+x^2\right) y_2+2 y y_1^2=0\)
- D \(\left(1+y^2\right) y_2-2 y y_1^2+y=0\)
Answer & Solution
Correct Answer
(B) \(\left(1+y^2\right) y_2-2 y y_1^2=0\)
Step-by-step Solution
Detailed explanation
\(y=\tan (a x+b)\) \(y_1=a \sec^2(a x+b) = a(1+y^2)\) \(a = \frac{y_1}{1+y^2}\) \(y_2=a \cdot (2y y_1)\) \(y_2 = \frac{y_1}{1+y^2} \cdot 2y y_1\) \((1+y^2) y_2 = 2y y_1^2\) \((1+y^2) y_2 - 2y y_1^2 = 0\)
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