Separation of variables: general and particular solutions Problem set B

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

1. (a) \(\dfrac{dy}{dx}=-3y\), \(y>0\)
2. (b) \(\dfrac{dy}{dx}=0.4(y-5)\), \(y>5\)
3. (c) \(\dfrac{dy}{dx}=2(9-y)\), \(y<9\)

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

1. (a) \(\dfrac{dy}{dx}=6xy^2\)
2. (b) \(\dfrac{dy}{dx}=\dfrac{5y}{x}\), \(x>0,\ y>0\)
3. (c) \(y\dfrac{dy}{dx}=7x^2\), \(y>0\)

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

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

Find the particular solution of each differential equation.

1. (a) \(\dfrac{dy}{dx}=5xy\), \(y=3\) when \(x=0\); give \(y\) in terms of \(x\).
2. (b) \(y^2\dfrac{dy}{dx}=6x\), \(y=2\) when \(x=1\); give \(y\) in terms of \(x\).

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

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

Given that \(\dfrac{dy}{dx}=\dfrac{y^2}{x^2}\), \(x>0\) and that \(y=0.5\) when \(x=1\), find \(y\) in terms of \(x\).