Differential Equations

1. Check this before you revise anything

"Formation of a differential equation" is gone. Older editions had a dedicated section — given a family of curves with $n$ arbitrary constants, differentiate $n$ times and eliminate the constants to build the differential equation they satisfy — complete with its own worked examples and its own exercise. The current book's section list is just:

A full-text search of the entire 38-page chapter for "formation," "eliminating arbitrary constants," or any "differentiate $n$ times" technique returns zero hits.

Across all 98 questions in this chapter's five exercises and Miscellaneous Exercise, not one asks you to build a differential equation from a given family of curves — every question either classifies an equation (order/degree), verifies a proposed solution, or solves one using separation, the homogeneous substitution, or the linear integrating-factor method. This matches the syllabus line exactly: "definition, order and degree, general and particular solutions... solution of differential equations."

The old stub taught formation as its own section, with the rule "differentiate as many times as there are constants; the number of arbitrary constants equals the order" — a real technique, just not one that exists in this edition. Removed entirely from this rebuild.

This is also a large chapter — 98 questions across five exercises plus a Miscellaneous Exercise. Several exercises hide extra questions past their own stated range: Exercise 9.1's instruction says "Exercises 1 to 10" but the exercise actually runs to Q12 (two unannounced MCQs on degree and order); Exercise 9.2 says "Exercises 1 to 10" but runs to Q12; Exercise 9.4 and 9.5 both add MCQs the same way.

None of these trailing MCQs are flagged by an explicit "Choose the correct answer in Exercises X and Y" instruction line the way they are in other chapters — they simply appear as directly-numbered questions with inline options.

2. What a Differential Equation Is (Textbook 9.1 to 9.2)

An ordinary algebraic equation such as $x^2-3x+2=0$ asks for a number. A differential equation asks for a function, and it does so by stating a relationship that the function and its derivatives must satisfy.

$\frac{dy}{dx}=2x\text{is a differential equation; its solutions are the functions }y=x^2+C$

The defining feature is that a derivative appears. An equation involving derivatives of a dependent variable with respect to a single independent variable is an ordinary differential equation. This chapter deals only with ordinary equations, and only with functions of one variable.

Notice immediately that the solution is not one function but a whole family — every value of $C$ gives a curve satisfying the equation. That plurality is the central fact of the chapter, and sections 9.3 onwards are largely about managing it.

Why this matters beyond the exam. Any statement of the form "the rate of change of $X$ is proportional to $X$" is a differential equation. Population growth, radioactive decay, Newton's law of cooling and the draining of a tank are all in Exercise 9.3 for exactly this reason.

3. Order and Degree (Textbook 9.2.1 to 9.2.2)

These two classifications are what the whole of Exercise 9.1 tests, and they are asked separately.

Order is the order of the highest derivative appearing in the equation. It requires no preparation of the equation — you simply look for the highest derivative present.

Note the last row: the highest derivative is $\frac{d^{3}y}{d{x}^3}$, so the order is 3. The power of 2 on it does not affect the order at all — that power is the degree.

Degree is the power of the highest-order derivative, after the equation has been cleared of radicals and fractions in its derivatives. Two conditions must be met before the question is even meaningful.

First, the equation must be expressible as a polynomial in its derivatives. Second, it must actually be cleared — an equation containing $\sqrt{\frac{dy}{dx}}$ has no degree until you square to remove the radical.

When degree is undefined. If the derivatives appear inside a transcendental function, no amount of rearranging produces a polynomial, and the degree does not exist:

$y^{\prime \prime \prime }+\sin (y^{\prime \prime })=0\text{has order }3\text{ and no defined degree}$

because $\sin (y^{\prime \prime })$ expands to an infinite series in $y^{\prime \prime }$. The same applies to $e^{y^{\prime }}$, $\log (y^{\prime })$ and similar terms. Exercise 9.1 includes several of these, and the expected answer is explicitly "degree not defined" — not zero, and not left blank.

4. General and Particular Solutions (Textbook 9.3)

A solution of a differential equation is a function that satisfies it identically when substituted in. Exercise 9.2 consists entirely of verifying that a proposed function is a solution, which is done by differentiating it the required number of times and substituting back.

General solution. The solution containing as many independent arbitrary constants as the order of the equation. A first-order equation has one constant, a second-order equation has two.

Geometrically the general solution is a family of curves. The equation $\frac{dy}{dx}=2x$ has general solution $y=x^2+C$, which is the set of all vertically-shifted parabolas.

Particular solution. Obtained by assigning specific values to every arbitrary constant, which picks out one curve from the family. A particular solution therefore contains zero arbitrary constants, whatever the order of the equation.

The values are fixed by extra information, usually an initial condition of the form $y=y_0$ when $x=x_0$. Substituting the condition into the general solution produces an equation in the constants, and solving it completes the problem.

The standard exam shape is: solve the equation to get the general solution, then apply the given condition to find the constant, then state the particular solution. Marks are allocated to all three steps, so write the general solution as an explicit intermediate line even when the question only asks for the particular one.

5. Variables Separable (Textbook 9.4.1)

The first and simplest of the three methods. It applies when the equation can be written with all the $y$ terms on one side and all the $x$ terms on the other.

The test. The equation has the form:

$\frac{dy}{dx}=g(x)h(y)$

that is, the right-hand side factorises into a function of $x$ alone times a function of $y$ alone.

The method. Divide through by $h(y)$, multiply by $dx$, and integrate each side with respect to its own variable:

$\frac{1}{h(y)}dy=g(x)dx⟹\int \frac{1}{h(y)}dy=\int g(x)dx+C$

Only one constant of integration is needed. Writing $C_1$ on the left and $C_2$ on the right and then combining them into a single $C$ is correct but wastes time.

A practical point on the constant. When both integrals produce logarithms, it is almost always cleaner to write the constant as $\log C$ rather than $C$. Then $\log y=\log x+\log C$ becomes $y=Cx$ directly, instead of $y=e^{C}x$ requiring a second renaming. Several Exercise 9.3 answers are stated in this form.

Word problems on growth, decay and cooling all reduce to this method, since "rate proportional to amount" gives $\frac{dy}{dt}=ky$, which separates immediately.

6. Homogeneous Differential Equations (Textbook 9.4.2)

The test. A function $F(x,y)$ is homogeneous of degree zero if:

$F(\lambda x,\lambda y)={\lambda }^{0}F(x,y)=F(x,y)$

for every non-zero $\lambda$. In practice this means every term in the expression has the same total degree in $x$ and $y$, so the scaling cancels. Examples include $\frac{x^2+y^2}{xy}$ and $\frac{y}{x}+\sin \frac{y}{x}$.

A differential equation $\frac{dy}{dx}=F(x,y)$ with such an $F$ is called homogeneous. Equivalently, the whole right-hand side can be written as a function of the single combination $y/x$.

The substitution. Put $y=vx$, where $v$ is a new function of $x$. Differentiating as a product:

$\frac{dy}{dx}=v+x\frac{dv}{dx}$

Substituting both into the equation turns it into a relation between $v$ and $x$ only — and that relation is always separable. Solve it by the method of the previous section, then replace $v$ by $y/x$ at the very end.

Do not forget the back-substitution. Leaving the answer in terms of $v$ is a common way to lose the final mark, since $v$ was never part of the original problem.

The mirror-image case. Sometimes the equation is more naturally homogeneous when read as $x$ in terms of $y$, that is $\frac{dx}{dy}=G(x,y)$. Then substitute $x=vy$ instead, giving $\frac{dx}{dy}=v+y\frac{dv}{dy}$, and proceed identically. Choosing the wrong direction still works in principle but usually produces a far messier integral, so inspect the equation before committing.

7. Linear Differential Equations (Textbook 9.4.3)

The standard form. A first-order equation is linear when it can be written as:

$\frac{dy}{dx}+Py=Q$

where $P$ and $Q$ are functions of $x$ alone, or constants. The defining feature is that $y$ and $\frac{dy}{dx}$ each appear to the first power and are not multiplied together.

The integrating factor. Define:

$\text{I.F.}=e^{\int Pdx}$

Multiplying the equation through by this factor makes the left-hand side an exact derivative — it becomes precisely $\frac{d}{dx}\left(y\cdot \text{I.F.}\right)$. That is the entire point of the construction, and it is why the method works.

Integrating both sides then gives the solution:

$y\cdot \text{I.F.}=\int \left(Q\cdot \text{I.F.}\right)dx+C$

Simplifying the integrating factor. The exponential and logarithm almost always collapse. If $P=2\tan x$ then $\int Pdx=2\log |\sec x|$, so $\text{I.F.}=e^{2\log |\sec x|}={\sec }^{2}x$. Leaving it as an unsimplified exponential makes the next integral far harder than it needs to be.

The mirror-image form. When isolating $y$ produces a messy equation but isolating $x$ does not, use:

$\frac{dx}{dy}+P_{1}x=Q_1,\text{I.F.}=e^{\int P_{1}dy}$

with $P_1,Q_1$ functions of $y$, solving for $x$ as a function of $y$. Exercise 9.5 questions 10 to 12 are built specifically to reward spotting this.

Worked, mirroring the textbook's own technique. Solve $\frac{dy}{dx}+2y\tan x=\sin x$, given $y=0$ when $x=\frac{\pi }{3}$.

This is linear with $P=2\tan x$ and $Q=\sin x$, so $\text{I.F.}=e^{\int 2\tan xdx}=e^{2\log |\sec x|}={\sec }^{2}x$.

Then $y{\sec }^{2}x=\int \sin x{\sec }^{2}xdx=\int \sec x\tan xdx=\sec x+C$, giving the general solution $y=\cos x+C{\cos }^{2}x$.

Substituting the condition $y\left(\frac{\pi }{3}\right)=0$: $0=\frac{1}{2}+C\cdot \frac{1}{4}$, so $C=-2$. The particular solution is $y=\cos x-2{\cos }^{2}x$.

Summary

• A differential equation relates a function to its derivatives, and its solution is a family of functions, not a single one.
• Order is the highest derivative present and needs no preparation of the equation; degree is the power of that highest derivative after clearing radicals and fractions.
• Degree is undefined whenever a derivative sits inside a transcendental function such as $\sin (y^{\prime \prime })$, $e^{y^{\prime }}$ or $\log (y^{\prime })$ — the expected answer is "not defined", not zero.
• The general solution carries as many arbitrary constants as the order and represents a family of curves; a particular solution fixes them all using an initial condition and carries none.
• "Formation of a differential equation" from a family of curves is not part of the current edition — every question here classifies, verifies, or solves an equation, never builds one.
• Variable separable: write $\frac{1}{h(y)}dy=g(x)dx$ and integrate each side; use $\log C$ as the constant when logarithms appear on both sides.
• Homogeneous: confirm $F(\lambda x,\lambda y)=F(x,y)$, substitute $y=vx$ with $\frac{dy}{dx}=v+x\frac{dv}{dx}$, solve the resulting separable equation, then substitute $v=y/x$ back.
• Use $x=vy$ instead whenever the equation is more naturally homogeneous in the other direction.
• Linear: $\frac{dy}{dx}+Py=Q$ gives $\text{I.F.}=e^{\int Pdx}$ and $y\cdot \text{I.F.}=\int (Q\cdot \text{I.F.})dx+C$; always simplify the integrating factor before integrating.
• Several exercises hide extra MCQs past their own stated question range, with no "choose the correct answer" lead-in line — always check the actual last page, not just the instruction sentence.