Forming a differential equation

What you will be able to do

• form the differential equation of a family of curves by eliminating the constants;
• say its order and degree from the number of constants and the form of the result;
• check that a family satisfies the equation you formed.

One equation for a whole family

Think of all the straight lines through the origin, $y = mx$. The number $m$ changes from line to line. Is there one equation that every one of these lines satisfies, with no $m$ in it?

Differentiate: $y' = m$. Put this in $y = mx$ and we get $y = xy'$. The constant has gone. Every line through the origin satisfies $x\,y' - y = 0$.

This is the idea of the lesson. A family with arbitrary constants becomes one differential equation, in which the constants have been eliminated.

The method

1. Count the constants. Say there are $n$ of them. The equation will have order $n$.
2. Differentiate the equation of the family $n$ times.
3. Eliminate the constants from these $n + 1$ equations.
4. Tidy up. Write the result without roots or fractions that hide a derivative, and then read the order and the degree.

The constants must be independent. In $y = (A + B)x$ the two letters make one constant, $A + B$, so the family has only one constant and the equation has order $1$.

Family with one constant

Example 1. The family $y = Cx^3$. Differentiate: $y' = 3Cx^2$. Then $C = \dfrac{y'}{3x^2}$ and also $C = \dfrac{y}{x^3}$. Equate: $\dfrac{y'}{3x^2} = \dfrac{y}{x^3}$, so $x\,y' = 3y$.

At $x = 1$ the curve $y = Cx^3$ has height $C$ and slope $3C$. So the slope is three times the height at $x = 1$, which is what $x\,y' = 3y$ says. Slide $C$ and watch the last reading.

Example 2. The parabolas $y^2 = 4ax$. Differentiate: $2yy' = 4a$. Put $4a = \dfrac{y^2}{x}$ from the family. Then $2yy' = \dfrac{y^2}{x}$, so $2xy' = y$ (for $y \ne 0$).

Example 3. Circles of radius $5$ with the centre on the $x$-axis: $(x - a)^2 + y^2 = 25$. Differentiate: $(x - a) + yy' = 0$, so $x - a = -yy'$. Put this in the family: $y^2y'^2 + y^2 = 25$. So $y^2\left(1 + y'^2\right) = 25$. The order is $1$ and the degree is $2$.

Family with two constants

Example 4. The family $y = Ae^x + Be^{2x}$ has two constants, so we differentiate twice.

• $y' = Ae^x + 2Be^{2x}$
• $y'' = Ae^x + 4Be^{2x}$

Subtract to remove $A$: $y' - y = Be^{2x}$ and $y'' - y' = 2Be^{2x}$. Then $y'' - y' = 2(y' - y)$, so

Example 5. The family $y = A\cos 2x + B\sin 2x$. Differentiate twice: $y'' = -4A\cos 2x - 4B\sin 2x = -4y$. So $y'' + 4y = 0$. Here the constants vanish at once.

Example 6. The family $y = ax + \dfrac{b}{x}$. Differentiate twice: $y' = a - \dfrac{b}{x^2}$ and $y'' = \dfrac{2b}{x^3}$. So $b = \dfrac{x^3y''}{2}$ and $a = y' + \dfrac{xy''}{2}$. Substitute: $y = xy' + \dfrac{x^2y''}{2} + \dfrac{x^2y''}{2}$, which gives $x^2y'' + xy' - y = 0$.

Practise with new numbers

Try it in Python

The program tests two families. It checks that $y = Cx^3$ satisfies $x\,y' = 3y$ for three values of $C$. Then it checks that $y = Ax + Bx^2$ satisfies $x^2y'' - 2xy' + 2y = 0$ for three pairs $(A, B)$.

Common mistakes

• Differentiating too few times. With two constants you need $y'$ and $y''$.
• Leaving a constant in the answer. The final equation must not contain $A$, $B$, $C$, $a$, $b$ or $m$.
• Counting dependent constants. $y = Ae^{x + B}$ is $y = (Ae^B)e^x$, with one constant.
• Forgetting to clear a root. Read the degree after the equation has no root of a derivative.
• Not checking. Put one member of the family back into your equation.

Remember these

• A family with $n$ independent constants gives an equation of order $n$.
• Differentiate $n$ times and eliminate the constants.
• Lines through the origin, $y = mx$, give $x\,y' = y$. The family $y = Cx^3$ gives $x\,y' = 3y$.
• Check the result with a member of the family.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. Form the differential equation of the family $y = Cx^2$.

2. Form the differential equation of the family $y = ae^{2x} + be^{-2x}$.

3. Form the differential equation of the family $y = ax + a^2$ and state its degree.

4. Form the differential equation of the family $y = e^x(A\cos x + B\sin x)$.

5. How many constants does the family of all circles in the plane have, and what is the order of its differential equation?

References

• OpenStax, Calculus Volume 2, Chapter 4, section 4.1 Basics of Differential Equations.
• NCERT, Mathematics Part II for Class XII, Chapter 9, the section on formation of a differential equation whose general solution is given.