Differential equations Worked Solutions — A-Level Maths

Fully worked, step-by-step solutions to A-Level Differential equations questions. See exactly how to solve problems on separation of variables, general solution, exponential, modelling.

separation of variablesgeneral solutionexponentialmodellingdecaycooling
A-Level70 questionsStep-by-step solutions
Question 1
2 markseasy
Find the general solution of the differential equation dydx=2y\frac{dy}{dx} = 2 y.

Worked solution

  1. Separate the variables

    1y dy=2 dx\frac{1}{y}\,dy = 2\,dx

    Collect all the y terms on one side and the x terms on the other.

  2. Integrate both sides

    ln⁡y=2x+c\ln y = 2 x + c

    Integrate each side with respect to its own variable and add one constant of integration.

  3. State the general solution

    y=Ae2xy = A e^{2 x}

    This is the general solution; the arbitrary constant absorbs ece^{c}.

Answer
y=Ae2xy = A e^{2 x}
Question 2
2 markseasy
Find the general solution of the differential equation dydx=3y\frac{dy}{dx} = 3 y.

Worked solution

  1. Separate the variables

    1y dy=3 dx\frac{1}{y}\,dy = 3\,dx

    Collect all the y terms on one side and the x terms on the other.

  2. Integrate both sides

    ln⁡y=3x+c\ln y = 3 x + c

    Integrate each side with respect to its own variable and add one constant of integration.

  3. State the general solution

    y=Ae3xy = A e^{3 x}

    This is the general solution; the arbitrary constant absorbs ece^{c}.

Answer
y=Ae3xy = A e^{3 x}
Question 3
2 markseasy
Find the general solution of the differential equation dydx=xy\frac{dy}{dx} = x y.

Worked solution

  1. Separate the variables

    1y dy=x dx\frac{1}{y}\,dy = x\,dx

    Collect all the y terms on one side and the x terms on the other.

  2. Integrate both sides

    ln⁡y=x22+c\ln y = \frac{x^{2}}{2} + c

    Integrate each side with respect to its own variable and add one constant of integration.

  3. State the general solution

    y=Aex22y = A e^{\frac{x^{2}}{2}}

    This is the general solution; the arbitrary constant absorbs ece^{c}.

Answer
y=Aex22y = A e^{\frac{x^{2}}{2}}
Question 4
2 markseasy
Find the general solution of the differential equation dydx=yx\frac{dy}{dx} = \frac{y}{x}.

Worked solution

  1. Separate the variables

    1y dy=1x dx\frac{1}{y}\,dy = \frac{1}{x}\,dx

    Collect all the y terms on one side and the x terms on the other.

  2. Integrate both sides

    ln⁡y=ln⁡(x)+c\ln y = \ln{\left(x \right)} + c

    Integrate each side with respect to its own variable and add one constant of integration.

  3. State the general solution

    y=Axy = A x

    This is the general solution; the arbitrary constant absorbs ece^{c}.

Answer
y=Axy = A x
Question 5
2 markseasy
Find the general solution of the differential equation dydx=ycos⁡(x)\frac{dy}{dx} = y \cos{\left(x \right)}.

Worked solution

  1. Separate the variables

    1y dy=cos⁡(x) dx\frac{1}{y}\,dy = \cos{\left(x \right)}\,dx

    Collect all the y terms on one side and the x terms on the other.

  2. Integrate both sides

    ln⁡y=sin⁡(x)+c\ln y = \sin{\left(x \right)} + c

    Integrate each side with respect to its own variable and add one constant of integration.

  3. State the general solution

    y=Aesin⁡(x)y = A e^{\sin{\left(x \right)}}

    This is the general solution; the arbitrary constant absorbs ece^{c}.

Answer
y=Aesin⁡(x)y = A e^{\sin{\left(x \right)}}

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