Differential Equations Class 12: notes and important questions

Differential Equations — revision notes

1. Order and degree

• Order: order of the highest derivative present.
• Degree: power of the highest-order derivative, after the equation is made a polynomial in the derivatives (clear radicals and fractional powers first). If a derivative sits inside \(\sin\), \(e^{(\,)}\), \(\ln\), etc., the degree is not defined.
• Order and degree (when defined) are always positive integers.

2. Solutions

General solution: contains as many arbitrary constants as the order. Particular solution: constants fixed by given conditions. To verify a solution, differentiate and substitute.

3. Variables separable: \(\dfrac{dy}{dx} = g(x)h(y)\)

\(\displaystyle\int \dfrac{dy}{h(y)} = \displaystyle\int g(x)\,dx + C\).

4. Homogeneous: \(\dfrac{dy}{dx} = F(x, y)\) with \(F(\lambda x, \lambda y) = F(x, y)\)

Put \(y = vx\), \(\dfrac{dy}{dx} = v + x\dfrac{dv}{dx}\); the equation becomes separable in \(v\) and \(x\). (If \(\dfrac{dx}{dy} = G(x, y)\) is more convenient, put \(x = vy\).) Replace \(v\) by \(\dfrac yx\) at the end.

5. Linear: \(\dfrac{dy}{dx} + Py = Q\) (\(P, Q\) functions of \(x\) or constants)

IF \(= e^{\int P\,dx}\); solution \(y \cdot \text{IF} = \displaystyle\int Q \cdot \text{IF}\,dx + C\). For \(\dfrac{dx}{dy} + P_1x = Q_1\): IF \(= e^{\int P_1\,dy}\), \(x \cdot \text{IF} = \displaystyle\int Q_1 \cdot \text{IF}\,dy + C\).

Worked example 1 (separable)

\(\dfrac{dy}{dx} = xy\): \(\ln|y| = \dfrac{x^2}{2} + c \Rightarrow y = Ce^{x^2/2}\).

Worked example 2 (linear)

\(x\dfrac{dy}{dx} + y = 3x^2\): \(\dfrac{d}{dx}(xy) = 3x^2 \Rightarrow xy = x^3 + C \Rightarrow y = x^2 + \dfrac Cx\).

Worked example 3 (homogeneous)

\(\dfrac{dy}{dx} = \dfrac yx + \dfrac xy\): \(y = vx \Rightarrow x\dfrac{dv}{dx} = \dfrac1v \Rightarrow \dfrac{v^2}{2} = \ln|x| + C \Rightarrow y^2 = 2x^2(\ln|x| + C)\).

Common errors

• Stating the degree without first clearing a radical or fractional power.
• Forgetting to write the equation in standard form (coefficient of \(\dfrac{dy}{dx}\) equal to \(1\)) before finding \(P\).
• Writing \(e^{\int P\,dx}\) with a “\(+C\)” inside — the IF needs no constant.
• Adding the constant only after dividing: in \(y \cdot \text{IF} = \int Q\cdot\text{IF}\,dx + C\), \(C\) must be divided by the IF too.
• Leaving the answer in \(v\) instead of \(\dfrac yx\).

Board-exam tips

• Say which type the equation is before solving (a step mark often depends on it).
• If \(\dfrac{dy}{dx}\) form is not linear in \(y\), try \(\dfrac{dx}{dy}\) — it may be linear in \(x\).
• In word problems, write the rate equation with units, and check the answer against the story (e.g. temperature approaches room temperature).