Differential Equations of Form dy/dx = f(x) or f(y)

Here you will learn how to find the general solution of differential equations of form dy/dx = f(x) or f(y) with examples.

Let’s begin –

Differential Equations of Form dy/dx = f(x) or f(y)

(1) Differential Equations of Form dy/dx = f(x)

To solve this type of differential equations we integrate both sides to obtain the general solution as discussed below.

We have. 

\(dy\over dx\) = f(x) \(\iff\) dy = f(x)dx

Integrating both sides, we obtain

\(\int \) dy = \(\int \) f(x) dx + C or,

y = \(\int \) f(x) dx + C, 

which gives general solution of the differential equation.

Example : Solve the given differential equation : \(dy\over dx\) = \(x\over x^2 + 1\)

Solution : We have,

\(dy\over dx\) = \(x\over x^2 + 1\)

\(\implies\) dy = \(x\over x^2 + 1\)dx

Integrating both sides, we get

\(\int \) dy = \(\int \) \(x\over x^2 + 1\)dx

\(\implies\) dy = \(1\over 2\) \(2x\over x^2 + 1\)dx

\(\implies\) y = \(1\over 2\) \(log|x^2 + 1|\) + C

Clearly, y = \(1\over 2\) \(log|x^2 + 1|\) + C is defined for all x \(\in\) R.

Hence, y = \(1\over 2\) \(log|x^2 + 1|\) + C, x \(\in\) R is the solution of the given differential equation.

(2) Differential Equations of Form dy/dx = f(y)

To solve this type of differential equations we integrate both sides to obtain the general solution as discussed below.

We have. 

\(dy\over dx\) = f(y) 

\(\implies\) \(dx\over dy\) = \(1\over f(y)\), provided that f(y) \(\ne\) 0

\(\implies\) dx = \(1\over f(y)\) dy

Integrating both sides, we obtain

\(\int \) dx = \(\int \) \(1\over f(y)\) dy + C or,

x = \(\int \) \(1\over f(y)\) dy + C, 

which gives general solution of the differential equation.

Example : Solve the given differential equation :  \(dy\over dx\) = \(1\over y^2 + sin y\)

Solution : We have,

\(dy\over dx\) = \(1\over y^2 + sin y\)

\(\implies\) \(dx\over dy\) = \(y^2 + sin y\)

\(\implies\) dx = \(y^2 + sin y\)dx

Integrating both sides, we get

\(\int \) dx = \(\int \) \(y^2 + sin y\)dx

\(\implies\) x = \(y^3\over 3\) – cosy + C

Hence, x = \(y^3\over 3\) – cosy + C is the solution of the given differential equation.

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