Solve the differential equation: . A particle falls from res

Question

Solve the differential equation: $\frac{dy}{dx} = (x + y)^2$ . A particle falls from rest in a medium in which the resistance is $\lambda v^2$ per unit mass, v being the velocity of the particle at time t. Prove that the distance fallen in time t is $\frac{1}{\lambda} \ln \cosh (t\sqrt{g\lambda})$ , where g is the acceleration due to gravity.

09 Jan 2026

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 __________ _________ ___ _________ ____ _________ ___ __________ ____ _____ ______.
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