Question

Choose the correct general solution from the given option for the homogeneous differential equation \[\dfrac{{dy}}{{dx}} = \sin x + 2x\].
A \[y = {x^2} - \cos x + c\]
B \[y = \cos x + {x^2} + c\]
C \[y = \cos x + 2 + c\]
D \[y = \cos x + 2 + c\]

Answer 1

Hint: The differential equation is a homogeneous differential equation. Hence to find a general solution, use variable separation methods. There are two variables x and y. Separate the x containing terms and y containing terms. Then integrate the equation to find a general solution of the differential equation.

Formula Used: \[\begin{array}{l}\int {\sin xdx} = \cos x + c\\\int {{x^n}dx} = \dfrac{{{x^{n + 1}}}}{{n + 1}} + c\end{array}\]
Here, c is an arbitrary constant and n can not be zero.

Complete step by step solution: The given differential equation is \[\dfrac{{dy}}{{dx}} = \sin x + 2x\].
First simplify the equation by separating the x and y variables. Here, multiply by \[dx\]on both sides of the equation.
\[dy = \left( {\sin x + 2x} \right)dx\]
Remove the bracket on the right side of the equation.
\[dy = \sin xdx + 2xdx\]
Now, integrate both sides of the equation.
 \[\begin{array}{l}\int {dy} = \int {\sin xdx} + \int {2xdx} \\y = \cos x + 2\int {xdx} \end{array}\]
Simplify the equation to find a general solution.
\[\begin{array}{l}y = \cos x + 2\dfrac{{{x^2}}}{2} + c\\y = \cos x + {x^2} + c\end{array}\]
Hence, the general solution is \[y = \cos x + {x^2} + c\] .

Option ‘A’ is correct

Note: The very common mistake while solving this kind of question is integration of \[\sin x\]is \[ - \cos x\]which is wrong. One more often mistake done here is integration of constant, which is arbitary need not to add it twice.