Question

In Problems \(1 - 6\), determine whether the given equation is separable, linear, neither, or both.

\({\bf{x}}\frac{{{\bf{dx}}}}{{{\bf{dt}}}}{\bf{ + x}}{{\bf{t}}^{\bf{2}}}{\bf{ = sin}}\;{\bf{t}}\).

Short answer

The given equation is a neither separable nor linear.

Step-by-step solution

General form of separable and linear equations

Separable equation: A first-order equation is separable if it can be written in the form \(\frac{{{\bf{dy}}}}{{{\bf{dx}}}}{\bf{ = g}}\left( {\bf{x}} \right){\bf{p}}\left( {\bf{y}} \right)\).

Linear equation: A first-order equation is linear if it can be written in the form\(\frac{{{\bf{dy}}}}{{{\bf{dx}}}}{\bf{ + P}}\left( {\bf{x}} \right){\bf{y = Q}}\left( {\bf{x}} \right)\).

Evaluate the given equation

Given that, \({\bf{x}}\frac{{{\bf{dx}}}}{{{\bf{dt}}}}{\bf{ + x}}{{\bf{t}}^{\bf{2}}}{\bf{ = sin}}\;{\bf{t}} \cdot \cdot \cdot \cdot \cdot \cdot \left( 1 \right)\)

Evaluative the given equation is separable or linear.

\(\begin{array}{c}{\bf{x}}\frac{{{\bf{dx}}}}{{{\bf{dt}}}}{\bf{ + x}}{{\bf{t}}^{\bf{2}}}{\bf{ = sin}}\;{\bf{t}}\\{\bf{x}}\frac{{{\bf{dx}}}}{{{\bf{dt}}}}{\bf{ = sin}}\;{\bf{t - x}}{{\bf{t}}^{\bf{2}}}\\\frac{{{\bf{dx}}}}{{{\bf{dt}}}}{\bf{ = }}\frac{{{\bf{sin}}\;{\bf{t - x}}{{\bf{t}}^{\bf{2}}}}}{{\bf{x}}}\\\frac{{{\bf{dx}}}}{{{\bf{dt}}}}{\bf{ + }}{{\bf{t}}^{\bf{2}}}{\bf{ = }}\frac{{{\bf{sin}}\;{\bf{t}}}}{{\bf{x}}}\end{array}\)

Identifying method

Where \(\frac{{{\bf{dx}}}}{{{\bf{dt}}}}{\bf{ + }}{{\bf{t}}^{\bf{2}}}{\bf{ = }}\frac{{{\bf{sin}}\;{\bf{t}}}}{{\bf{x}}}\) and \(\frac{{{\bf{dx}}}}{{{\bf{dt}}}}{\bf{ = }}\frac{{{\bf{sin}}\;{\bf{t - x}}{{\bf{t}}^{\bf{2}}}}}{{\bf{x}}}\).

So, the given equation is not linear because of the term \(\frac{{{\bf{sin}}\;{\bf{t}}}}{{\bf{x}}}\) and referring to evaluation the equation of right-hand side we can’t found any product of function which depends only on x and function which depends only on t, so this equation is not separable.

Hence, the given equation is a neither separable nor linear.