Animated Solution for Mathematics - Differential Equations: The order of the differential equation whose general solution is given by $y=(C_1+C_2)\cos (x+C_3)-C_4{e}^{x+C_5}$, where $C_1,C_2,C_3,C_4,C_5$ are arbitrary constants, is

Unmasking the Differential Equation

Imagine you are standing before a massive, intimidating wall of algebra. The equation $y=(C_1+C_2)\cos (x+C_3)-C_4{e}^{x+C_5}$ stares back at you, flaunting five different constants.

Your instinct might be to panic and count them all, concluding that the order of the differential equation must be five. But hold on—in the world of JEE Advanced, appearances are often designed to deceive.

The core of this problem lies in the definition of the order of a differential equation: it is the number of independent arbitrary constants. Our goal is to strip away the facade and reveal the true, irreducible number of constants hiding underneath.

The Additive Deception

Let us look at the first term: $(C_1+C_2)\cos (x+C_3)$. You see two constants, $C_1$ and $C_2$, being added together.

In the realm of arbitrary constants, the sum of two unknowns is simply another unknown. If $C_1$ is any real number and $C_2$ is any real number, then $A=C_1+C_2$ is also just an arbitrary constant.

By substituting $A=C_1+C_2$, we have reduced our count from five to four. The equation now looks like:

The Exponential Trap

Now, turn your attention to the second term: $-C_4{e}^{x+C_5}$. This is where many students stumble.

Recall the fundamental law of exponents: $e^{a+b}=e^a\cdot e^b$. Applying this to our term, we get $e^{x+C_5}=e^x\cdot e^{C_5}$.

Now, look at the constants attached to the variable $e^x$. We have $C_4$ multiplied by $e^{C_5}$. Since $C_4$ and $C_5$ are arbitrary, the product $B=C_4\cdot e^{C_5}$ is, once again, just a single, new arbitrary constant.

With this substitution, our equation transforms into:

The Grand Reveal

We have successfully reduced the expression to $y=A\cos (x+C_3)-B{e}^x$. Let us expand the cosine term using the trigonometric identity:

Substituting this back, we get:

This simplifies to:

Let $K_1=A\cos C_3$, $K_2=-A\sin C_3$, and $K_3=-B$. The equation is now:

We have three independent constants: $K_1,K_2,$ and $K_3$. Since there are exactly three independent constants, the order of the differential equation is 3.

It is a beautiful example of how mathematical simplification reveals the underlying structure of a problem. Never let the initial complexity intimidate you; always look for the hidden simplicity.