First-order differential equations Worked Solutions — Further Maths Maths

Fully worked, step-by-step solutions to Further Maths First-order differential equations questions. See exactly how to solve problems on integrating-factor, linear-first-order, separable, first-order.

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Further Maths70 questionsStep-by-step solutions

Question 1

2 markseasy

Find an integrating factor for the differential equation

\frac{dy}{dx}+3y=\mathrm{e}^{2x}

Worked solution

Identify $P\left(x\right)$ and $Q\left(x\right)$
$P\left(x\right)=3,\qquad Q\left(x\right)=\mathrm{e}^{2x}$
$P$ is the coefficient of $y$ once the equation is in standard form.
Form the integrating factor
$\mu=\mathrm{e}^{\int 3\,dx}=\mathrm{e}^{3x}$
This is the multiplier that makes the left-hand side exact.
State the integrating factor
$\mu=\mathrm{e}^{3x}$
Multiplying the equation by this factor would make the left-hand side an exact derivative.

Answer

\mu=\mathrm{e}^{3x}

Question 2

2 markseasy

Find an integrating factor for the differential equation

\frac{dy}{dx}+\frac{2y}{x}=x^{3}

Worked solution

Identify $P\left(x\right)$ and $Q\left(x\right)$
$P\left(x\right)=\frac{2}{x},\qquad Q\left(x\right)=x^{3}$
$P$ is the coefficient of $y$ once the equation is in standard form.
Integrate $P\left(x\right)$
$\int \frac{2}{x}\,dx=2\ln{\left(x\right)}$
No constant of integration is needed at this stage.
Form the integrating factor
$\mu=\mathrm{e}^{\int \frac{2}{x}\,dx}=\mathrm{e}^{2\ln{\left(x\right)}}=x^{2}$
This is the multiplier that makes the left-hand side exact.
State the integrating factor
$\mu=x^{2}$
Multiplying the equation by this factor would make the left-hand side an exact derivative.

Answer

\mu=x^{2}

Question 3

2 markseasy

Find an integrating factor for the differential equation

\frac{dy}{dx}-4y=5

Worked solution

Identify $P\left(x\right)$ and $Q\left(x\right)$
$P\left(x\right)=-4,\qquad Q\left(x\right)=5$
$P$ is the coefficient of $y$ once the equation is in standard form.
Form the integrating factor
$\mu=\mathrm{e}^{\int -4\,dx}=\mathrm{e}^{-4x}$
This is the multiplier that makes the left-hand side exact.
State the integrating factor
$\mu=\mathrm{e}^{-4x}$
Multiplying the equation by this factor would make the left-hand side an exact derivative.

Answer

\mu=\mathrm{e}^{-4x}

Question 4

2 markseasy

Find an integrating factor for the differential equation

\frac{dy}{dx}+y\tan{\left(x\right)}=\cos{\left(x\right)}

Worked solution

Identify $P\left(x\right)$ and $Q\left(x\right)$
$P\left(x\right)=\tan{\left(x\right)},\qquad Q\left(x\right)=\cos{\left(x\right)}$
$P$ is the coefficient of $y$ once the equation is in standard form.
Integrate $P\left(x\right)$
$\int \tan{\left(x\right)}\,dx=-\ln{\left(\cos{\left(x\right)}\right)}$
No constant of integration is needed at this stage.
Form the integrating factor
$\mu=\mathrm{e}^{\int \tan{\left(x\right)}\,dx}=\mathrm{e}^{-\ln{\left(\cos{\left(x\right)}\right)}}=\sec{\left(x\right)}$
This is the multiplier that makes the left-hand side exact.
State the integrating factor
$\mu=\sec{\left(x\right)}$
Multiplying the equation by this factor would make the left-hand side an exact derivative.

Answer

\mu=\sec{\left(x\right)}

Question 5

2 markseasy

Find an integrating factor for the differential equation

\frac{dy}{dx}+\frac{5y}{x}=\frac{1}{x}

Worked solution

Identify $P\left(x\right)$ and $Q\left(x\right)$
$P\left(x\right)=\frac{5}{x},\qquad Q\left(x\right)=\frac{1}{x}$
$P$ is the coefficient of $y$ once the equation is in standard form.
Form the integrating factor
$\mu=\mathrm{e}^{\int \frac{5}{x}\,dx}=\mathrm{e}^{5\ln{\left(x\right)}}=x^{5}$
This is the multiplier that makes the left-hand side exact.
State the integrating factor
$\mu=x^{5}$
Multiplying the equation by this factor would make the left-hand side an exact derivative.

Answer

\mu=x^{5}

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