Given f(1)=1/3 and 3∫₁ˣf(t)dt=xf(x)−x³/3, find f(e) by differentiating both sides to get a linear differential equation and solving it with an integrating factor.
For real α, β let y_{α,β}(x) solve dy/dx+αy=xe^(βx), y(1)=1. S is the set of all such functions. Which given options belong to S? Multiple correct.
The solution curve y=y(x) of dy/dx − 3x⁵tan⁻¹(x³)y/(1+x⁶)^(3/2) = 2x·exp[(x³−tan⁻¹x³)/√(1+x⁶)] passes through the origin. Find y(1).
Solve the differential equation cos x dy = y(sin x − y)dx for x in (0,π/2), by recognising it as a Bernoulli equation and substituting t=1/y.
Solve the differential equation y=(x−y dx/dy)sin(x/y) with x(1)=π/2, using the substitution v=x/y to reduce it to a separable equation.