The variables x and y satisfy the differential equation
\(x^2 \frac{dy}{dx} + y^2 + y = 0\).
It is given that \(x = 1\) when \(y = 1\).
(a) Solve the differential equation to obtain an expression for y in terms of x.
(b) State what happens to the value of y when x tends to infinity. Give your answer in an exact form.
(i) Express \(\frac{100}{x^2(10-x)}\) in partial fractions.
(ii) Given that \(x = 1\) when \(t = 0\), solve the differential equation \(\frac{dx}{dt} = \frac{1}{100}x^2(10-x)\), obtaining an expression for \(t\) in terms of \(x\).
(i) Using partial fractions, find \(\int \frac{1}{y(4-y)} \, dy\).
(ii) Given that \(y = 1\) when \(x = 0\), solve the differential equation \(\frac{dy}{dx} = y(4-y)\), obtaining an expression for \(y\) in terms of \(x\).
(iii) State what happens to the value of \(y\) if \(x\) becomes very large and positive.
The variables x and y satisfy the differential equation
\((x + 1)(3x + 1) \frac{dy}{dx} = y,\)
and it is given that \(y = 1\) when \(x = 1\).
Solve the differential equation and find the exact value of \(y\) when \(x = 3\), giving your answer in a simplified form.
The variables x and t satisfy the differential equation \(\frac{dx}{dt} = x^2(1 + 2x)\), and \(x = 1\) when \(t = 0\).
Using partial fractions, solve the differential equation, obtaining an expression for t in terms of x.
The variables x and y satisfy the differential equation
\(\frac{dy}{dx} = \frac{y-1}{(x+1)(x+3)}\).
It is given that \(y = 2\) when \(x = 0\).
Solve the differential equation, obtaining an expression for \(y\) in terms of \(x\).
(i) Express \(\frac{1}{4-y^2}\) in partial fractions.
(ii) The variables \(x\) and \(y\) satisfy the differential equation \(\frac{dy}{dx} = \frac{x}{4-y^2}\), and \(y = 1\) when \(x = 1\). Solve the differential equation, obtaining an expression for \(y\) in terms of \(x\).
(i) Express \(\frac{1}{x(2x+3)}\) in partial fractions.
(ii) The variables \(x\) and \(y\) satisfy the differential equation \(x(2x+3) \frac{dy}{dx} = y\), and it is given that \(y = 1\) when \(x = 1\). Solve the differential equation and calculate the value of \(y\) when \(x = 9\), giving your answer correct to 3 significant figures.
Given that \(y = 1\) when \(x = 0\), solve the differential equation \(\frac{dy}{dx} = 4x(3y^2 + 10y + 3)\), obtaining an expression for \(y\) in terms of \(x\).
(i) Express \(\frac{1}{x^2(2x+1)}\) in the form \(\frac{A}{x^2} + \frac{B}{x} + \frac{C}{2x+1}\).
(ii) The variables \(x\) and \(y\) satisfy the differential equation \(y = x^2(2x+1) \frac{dy}{dx}\), and \(y = 1\) when \(x = 1\). Solve the differential equation and find the exact value of \(y\) when \(x = 2\). Give your value of \(y\) in a form not involving logarithms.
The variables x and y are related by the differential equation \(x \frac{dy}{dx} = 1 - y^2\).
When \(x = 2, y = 0\). Solve the differential equation, obtaining an expression for y in terms of x.
The variables x and y satisfy the differential equation \(\frac{dy}{dx} = xe^{y-x}\), and \(y = 0\) when \(x = 0\).
(a) Solve the differential equation, obtaining an expression for y in terms of x.
(b) Find the value of y when \(x = 1\), giving your answer in the form \(a - \ln b\), where a and b are integers.
The variables x and y satisfy the differential equation
\(e^{2x} \frac{dy}{dx} = 4xy^2\),
and it is given that \(y = 1\) when \(x = 0\).
Solve the differential equation, obtaining an expression for y in terms of x.
The variables x and y satisfy the differential equation \(\frac{dy}{dx} = xe^{x+y}\), and it is given that \(y = 0\) when \(x = 0\).
The variables \(x\) and \(y\) are related by the differential equation \[ \frac{dy}{dx} = \frac{1}{5}x y^{\frac{1}{2}} \sin\left(\frac{1}{3}x\right). \]
(i) Find the general solution, giving \(y\) in terms of \(x\).
(ii) Given that \(y = 100\) when \(x = 0\), find the value of \(y\) when \(x = 25\).
The variables x and y are related by the differential equation \(\frac{dy}{dx} = \frac{6xe^{3x}}{y^2}\).
It is given that \(y = 2\) when \(x = 0\). Solve the differential equation and hence find the value of \(y\) when \(x = 0.5\), giving your answer correct to 2 decimal places.
The variables x and y satisfy the differential equation
\(e^{4x} \frac{dy}{dx} = \cos^2 3y\).
It is given that \(y = 0\) when \(x = 2\).
Solve the differential equation, obtaining an expression for y in terms of x.
The variables x and y satisfy the differential equation \(\frac{dy}{dx} = \frac{1 + 4y^2}{e^x}\).
It is given that \(y = 0\) when \(x = 1\).
(a) Solve the differential equation, obtaining an expression for y in terms of x.
(b) State what happens to the value of y as x tends to infinity.
The variables x and ฮธ satisfy the differential equation
\(\sin \frac{1}{2} \theta \frac{dx}{d\theta} = (x + 2) \cos \frac{1}{2} \theta\)
for \(0 < \theta < \pi\). It is given that \(x = 1\) when \(\theta = \frac{1}{3} \pi\). Solve the differential equation and obtain an expression for \(x\) in terms of \(\cos \theta\).
(i) Differentiate \(\frac{1}{\sin^2 \theta}\) with respect to \(\theta\).
(ii) The variables \(x\) and \(\theta\) satisfy the differential equation \(x \tan \theta \frac{dx}{d\theta} + \csc^2 \theta = 0\), for \(0 < \theta < \frac{1}{2}\pi\) and \(x > 0\). It is given that \(x = 4\) when \(\theta = \frac{1}{6}\pi\). Solve the differential equation, obtaining an expression for \(x\) in terms of \(\theta\).