निम्नलिखित सीमा मान समस्या को हल कीजिए
इस सीमा मान समस्या को हल करने के लिए हम इसे सामान्यीकरण और रिक्त स्थानों में विभाजित करेंगे। यह समस्या एक सामान्य प्रकार की गर्मी की समस्याओं (Heat Equation) से संबंधित है, जिसमें समय के साथ स्थान में तापमान के वितरण का अध्ययन किया जाता है। समस्या है: \[ u_t = u_{xx}, \quad 0 < x _____ _______ ___ ____ ___ _______ ___ _______ _________ _________ ____ _____.
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If ƒ and g are arbitrary functions of their respective arguments, show that is a solution of
where,
Find the orthogonal trajectories of the family of parabolas x = cy2 and sketch their graph.
अवकल समीकरण का व्यापक हल प्राप्त कीजिए।
प्राचल विचार विधि से गुणांक
को हल 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.
Solve the IVP :
Find the general solution of the DE
y² ln y = x py + p² Does the equation has any singular solution? If yes, obtain it.