9) If the general solution of some differential equation is $$ y=a_{1}\left(a_{2}+a_{3} ight) \cdot \cos \left(x+a_{4} ight)-a_{3} e^{x+a_{0}} $$ then order of differential equation is $$ \begin{array}{ll} ext { (A) } 5 & ext { (B) } 4 \ ext { (C) } 6 & ext { (D) } 3\end{array} $$

Question

9) If the general solution of some differential equation is $$ y=a_{1}\left(a_{2}+a_{3}ight) \cdot \cos \left(x+a_{4}ight)-a_{3} e^{x+a_{0}} $$ then order of differential equation is $$ \begin{array}{ll}	ext { (A) } 5 & 	ext { (B) } 4 \ 	ext { (C) } 6 & 	ext { (D) } 3\end{array} $$

UpStudy AI · Accepted Answer

We are given
$$
y = a_{1}(a_{2}+a_{3})\cos(x+a_{4}) - a_{3}e^{x+a_{0}}.
$$
At first glance, there appear to be five arbitrary constants: $$a_0, a_1, a_2, a_3,$$ and $$a_4$$. However, we must check if all of these are independent or if some can be combined.

**Step 1. Rewrite the expression by combining constants where possible.**

Notice that the term $$a_{1}(a_{2}+a_{3})$$ multiplies $$\cos(x+a_4)$$. Introduce a new constant:
$$
A = a_{1}(a_{2}+a_{3}).
$$
Then the solution becomes:
$$
y = A \cos(x+a_{4}) - a_{3}e^{x+a_{0}}.
$$

Similarly, the exponential term can be written as:
$$
a_{3}e^{x+a_{0}} = a_{3}e^{a_{0}}e^x.
$$
Introduce another constant:
$$
B = a_{3}e^{a_{0}}.
$$
Thus the general solution simplifies to:
$$
y = A \cos(x+a_{4}) - B e^x.
$$

**Step 2. Count the independent arbitrary constants.**

In the rewritten form, the arbitrary constants are:
- $$A$$ (which is arbitrary because $$a_1$$ and $$a_2+a_3$$ can be chosen to produce any value),
- $$a_4$$ (the phase shift in the cosine),
- $$B$$ (the coefficient of $$e^x$$).

There are no other independent constants. Therefore, the total number of independent arbitrary constants is 3.

**Step 3. Conclude the order of the differential equation.**

The order of a differential equation is equal to the number of independent arbitrary constants in its general solution. Since we have 3 independent arbitrary constants, the order of the differential equation is

$$
\boxed{3}.
$$

Thus, the correct answer is (D) 3.
The differential equation is of order 3.

9) If the general solution of some differential equation is \( y=a_{1}\left(a_{2}+a_{3}\right) \cdot \cos \left(x+a_{4}\right)-a_{3} e^{x+a_{0}} \) then order of differential equation is \( \begin{array}{ll}\text { (A) } 5 & \text { (B) } 4 \\ \text { (C) } 6 & \text { (D) } 3\end{array} \)

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