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
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 ________ _____ ____ __________ _________ _______.
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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 ________ _____ ____ __________ _________ _______.
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Related Questions
in
is a linear homogeneous equation
State whether the following statements are true or false. Justify your answer with the help of a short proof or a counter example:
a) y2 is an integrating factor of the differential equation:
.
हल कीजिए :
Using the method of undetermined coefficients, solve the differential equation:
Solve: .
चार्पिट विधि द्वारा समीकरण का हल प्राप्त कीजिए।
Solve: .
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