JEE Mathematics 2024 Question with Solution

Given: $y(0)=0$ and the solution concludes that for $x=\frac{1}{2}$, we get $y\left(\frac{1}{2}\right)=\frac{54}{43}$, so $m+n=97$.

Find: the value of $m+n$.

Rewrite the differential equation in linear form as shown in the solution:

Thus,

and the integrating factor is

Multiplying by the integrating factor,

Integrating,

Using $y(0)=0$, the constant of integration is taken as $0$ in the solution. Then substituting $x=\frac{1}{2}$, the extracted solution states

Hence, $m=54$ and $n=43$, so

Therefore, the required numerical value is $97$.

The solution is internally inconsistent in its intermediate expressions. One approach states $y\left(\frac{1}{2}\right)=\frac{54}{43}$, while another approach mentions values equivalent to $65$ and $32$, yet both conclude $m+n=97$. Since the solution also gives Correct Answer: $97$, the final answer is taken as $97$.

Therefore, the required numerical value is $97$.