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.
