Separation of variables: general and particular solutions Problem set A

Find the general solution of each differential equation. Where the question gives the sign of \(y\), use it to remove the modulus.

1. (a) \(\dfrac{dy}{dx}=5y\), \(y>0\)
2. (b) \(\dfrac{dy}{dx}=0.2(y-6)\), \(y>6\)
3. (c) \(\dfrac{dy}{dx}=4-y\), \(y<4\)

Find the general solution of each differential equation. Give \(y\) in terms of \(x\).

1. (a) \(\dfrac{dy}{dx}=4xy^2\)
2. (b) \(\dfrac{dy}{dx}=\dfrac{3y}{x}\), \(x>0,\ y>0\)
3. (c) \(y\dfrac{dy}{dx}=5x^2\), \(y>0\)

Find the general solution of each differential equation. Give \(y\) in terms of \(x\).

1. (a) \(\dfrac{dy}{dx}=e^{-y}\sin x\)
2. (b) \(\dfrac{dy}{dx}=ye^{2x}\), \(y>0\)

Find the particular solution of each differential equation.

1. (a) \(\dfrac{dy}{dx}=3xy\), \(y=2\) when \(x=0\); give \(y\) in terms of \(x\).
2. (b) \(y^2\dfrac{dy}{dx}=4x\), \(y=3\) when \(x=1\); give \(y\) in terms of \(x\).

Given that \(\dfrac{dy}{dx}=x(y-1)\) and that \(y=4\) when \(x=0\), show that \(y=1+3e^{x^2/2}\).

Given that \(\dfrac{dy}{dx}=3x^2\cos^2y\), use separation of variables to show that \(\tan y=x^3+c\) for some constant \(c\).

Given that \(\dfrac{dy}{dx}=\dfrac{y^2}{x^2}\), \(x>0\) and that \(y=1\) when \(x=2\), find \(y\) in terms of \(x\).