Question
Solve the differential equation: . A particle falls from rest in a medium in which the resistance is
per unit mass, v being the velocity of the particle at time t. Prove that the distance fallen in time t is
, where g is the acceleration due to gravity.
Answer :
Word Count : 336
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Skip in 10
Let's solve the two problems step by step manually. --- 1. Solve the differential equation: [ \frac{dy}{dx} = (x + y)^2 ] Substitute: (v = x + y \implies y = v - x). Then [ \frac{dy}{dx} = \frac{dv}{dx} - 1 ] Substitute into the DE: [ \frac{dv}{dx} - 1 = v^2 \implies \frac{dv}{dx} = v^2 + 1 ] Separate variables: [ \frac{dv}{v^2 + 1} = dx ] Integrate both sides: [ \int \frac{dv}{v^2 + 1} = \int dx \implies \arctan(v) = x + C ] Substitute back (v = x
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Related Questions
Solve: .
बताइए कि निम्नलिखित कथन सत्य हैं या असत्य। संक्षिप्त उपपत्ति अथवा प्रत्युदाहरण की सहायता से अपने उत्तर की पुष्टि कीजिए :
क) y2 अवकल समीकरण :$
का समाकलन गुणक है।
ख) अवकल समीकरण , जहाँ
, के हल का अस्तित्व है लेकिन हल अद्वितीय नहीं है।
ग) अंतराल में समीकरण :
समघात रैखिक समीकरण है।
घ) अवकल समीकरण$
के हल का अस्तित्व है, परंतु हल अद्वितीय नहीं है।
ङ) फैफियन अवकल समीकरण समाकलनीय है।
हल कीजिए :
Solve: .
अवकल समीकरण :
$
को प्राचल विचरण विधि से हल कीजिए।
हल कीजिए : .
) हल कीजिए :