09Class 12 Maths

Differential Equations formulas

Order, degree and solution methods for Differential Equations. First identify the type, and the method follows.

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Types and solutions

Variable separable: $\displaystyle \frac{dy}{dx} = f(x)g(y) \;\Rightarrow\; \int\frac{dy}{g(y)} = \int f(x)\,dx$
Homogeneous: $\displaystyle \frac{dy}{dx} = F\left(\frac{y}{x}\right), \quad \text{put } y = vx$
Linear form: $\displaystyle \frac{dy}{dx} + Py = Q$
P and Q are functions of x.
Integrating factor: $\displaystyle \text{I.F.} = e^{\int P\,dx}$
Solution of the linear form: $\displaystyle y\cdot\text{I.F.} = \int Q\cdot\text{I.F.}\,dx + C$

Reporting the degree before clearing fractions and radicals of derivatives. The equation must be polynomial in its derivatives.
Forgetting the constant of integration, so the particular solution can't be found.
Using the integrating factor before rewriting the equation so dy/dx has coefficient 1.

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