Find the orthogonal trajectories of the family of parabolas x = cy2 and sketch their graph.
The given family of parabolas is \( x = c y^2 \). Differentiating both sides with respect to \( x \),
\[
\frac{dx}{dx} = c \cdot 2y \frac{dy}{dx}
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Solve the differential equation
where
Solve :
Solve the following boundary value problem
जैकोबी विधि से समीकरण
का पूर्ण समाकल ज्ञात कीजिए।
Classify the following statements as true or false. Give a short proof of a counter example in support of your answer.
i) The solution of the differential equation exists, but is not unique.
ii) The differential equation representing all tangents ty = x + t2 at the point (t², 2t) to the parabola y² = 4x is x(y')² + yy'+1=0
iii) The p.d.e. auxx + 2b uxy + cu,yy = 0 where a, are constants is irreducible when b2 ac = 0
iv) The functions f1(x) = cos²x, f2(x) = sin²x, f3(x) = sec² x and f4(x) = tan²x are linearly dependent on the interval ] - π/2, π/2 [
v) The solution of the second order partial differential equation involves two arbitrary constants.
समीकरण p2 +q2-2px-2qy+1 = 0 का पूर्ण समाकल ज्ञात कीजिए।
अवकल समीकरण का व्यापक हल प्राप्त कीजिए।