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Variable-Separable Differential Equations

A first-order differential equation is variable separable if it can be arranged so that all terms involving y and dy are on one side and all terms involving x and dx are on the other side.

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Student-friendly explanation

The standard idea is to rewrite dy/dx = f(x)g(y) as dy/g(y) = f(x) dx, then integrate both sides. The constant of integration is essential because the solution represents a family of curves.

How to write this in exams

  1. 1

    Start with the exact idea

    A first-order differential equation is variable separable if it can be arranged so that all terms involving y and dy are on one side and all terms involving x and dx are on the other side.

  2. 2

    Then show how to use it

    Rewrite the equation in differential form if useful. Move all y terms with dy to one side. Move all x terms with dx to the other side. Integrate both sides. Add the arbitrary constant. Simplify only if it preserves the solution correctly.

  3. 3

    Add one concrete example

    For dy/dx = xy, separate variables: dy/y = x dx. Integrate: ∫dy/y = ∫x dx, so log|y| = x2/2 + C.

  4. 4

    Avoid this incomplete answer

    Forgetting + C after integration, or writing ∫dy/(1 + y2) as log(1 + y2) instead of tan^(-1)y.

Definition

A first-order differential equation is variable separable if it can be arranged so that all terms involving y and dy are on one side and all terms involving x and dx are on the other side.

Example

For dy/dx = xy, separate variables: dy/y = x dx. Integrate: ∫dy/y = ∫x dx, so log|y| = x2/2 + C.

Rule to remember

Key form: dy/dx = f(x)g(y). Separation gives dy/g(y) = f(x) dx, provided g(y) is not zero on the interval considered. Then ∫dy/g(y) = ∫f(x) dx + C.

Memory hook

Separate first, integrate second; never integrate a mixed side.

Examples and method

Worked example

Example: Solve dy/dx = x(1 + y2). Separate: dy/(1 + y2) = x dx. Integrate both sides: tan^(-1)y = x2/2 + C. Hence the general solution is tan^(-1)y - x2/2 = C.

Method to apply

Rewrite the equation in differential form if useful. Move all y terms with dy to one side. Move all x terms with dx to the other side. Integrate both sides. Add the arbitrary constant. Simplify only if it preserves the solution correctly.

Diagram support

A diagram is not required because the solution method is symbolic. A slope-field visual may help conceptually, but CBSE solving questions usually require algebraic separation and integration.

How CBSE asks it

Usually asked as a direct solving problem, sometimes after rearrangement. The equation may appear as M(y)dy = N(x)dx or dy/dx = f(x)g(y).

Avoid common mistakes

Common confusion

Students often integrate dy/dx = f(x)g(y) directly with respect to x without separating y terms correctly, which treats y like a constant.

Common wrong answer

Forgetting + C after integration, or writing ∫dy/(1 + y2) as log(1 + y2) instead of tan^(-1)y.

Exam tip

After separating variables, check that one side contains only y with dy and the other contains only x with dx before integrating.

Quick check

Solve the separated form dy/y = 3x2 dx.

Integrating gives log|y| = x3 + C, or y = Ce^(x3) where C is a non-zero arbitrary constant; y = 0 may also satisfy the original form if allowed.

Answer writing and exam use

1-mark answer

A first-order differential equation is variable separable if it can be arranged so that all terms involving y and dy are on one side and all terms involving x and dx are on the other side.

2-mark answer

A first-order differential equation is variable separable if it can be arranged so that all terms involving y and dy are on one side and all terms involving x and dx are on the other side. Key form: dy/dx = f(x)g(y). Separation gives dy/g(y) = f(x) dx, provided g(y) is not zero on the interval considered. Then ∫dy/g(y) = ∫f(x) dx + C. For dy/dx = xy, separate variables: dy/y = x dx. Integrate: ∫dy/y = ∫x dx, so log|y| = x2/2 + C.

3-mark answer

The standard idea is to rewrite dy/dx = f(x)g(y) as dy/g(y) = f(x) dx, then integrate both sides. The constant of integration is essential because the solution represents a family of curves. Key form: dy/dx = f(x)g(y). Separation gives dy/g(y) = f(x) dx, provided g(y) is not zero on the interval considered. Then ∫dy/g(y) = ∫f(x) dx + C. Example: Solve dy/dx = x(1 + y2). Separate: dy/(1 + y2) = x dx. Integrate both sides: tan^(-1)y = x2/2 + C. Hence the general solution is tan^(-1)y - x2/2 = C. Usually asked as a direct solving problem, sometimes after rearrangement. The equation may appear as M(y)dy = N(x)dx or dy/dx = f(x)g(y). Forgetting + C after integration, or writing ∫dy/(1 + y2) as log(1 + y2) instead of tan^(-1)y.
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