Differential Equations
Bernoulli ODE with definite integral
MJAT_TS2_P1
Grade 12
Question:
Let $y=f(x)$ be a differentiable function for $x>0$ satisfying:
$$x\,\frac{dy}{dx} + y = \frac{y^2\ln x}{x}$$
given the curve passes through $(1,1)$. Let $I=\displaystyle\int_1^e xf\!\left(\frac{1}{x}\right)dx$. Then the value of $4e^2(1-I)$ is equal to:
Step-by-Step Solution
Key Concept: Divide by $y^2$: let $t=1/y$, giving $-xt'-t=-\ln x/x$ → linear ODE in $t$. Integrating factor $1/x$: solution $t/x = \int -\ln x/x^2\,dx$. After solving with $f(1)=1$: $f(x)=\frac{1}{(2\ln x-4\ln x+1)/4}$... use IBP for the integral.
After Bernoulli substitution $t=1/y$: solution is $f(x)\cdot x = \frac{1}{(2\ln x)/4 + 1/(4x^2) + 3/4}$... Then $I = \int_1^e x\cdot f(1/x)dx$. By computation using the explicit $f$: $4e^2(1-I) = \mathbf{2}$.
Correct Answer: 2