Mathematics - Continuity - Differentiability Question with Solution | TestHub

The given differential equation is $f(x)f^{\prime }(x)=x$. This can be written as $f(x)\frac{df}{dx}=x$. Integrating both sides with respect to $x$: $$\int f(x)\frac{df}{dx}dx=\int xdx$$ Let $u=f(x)$, then $du=f^{\prime }(x)dx$. The integral becomes $$\int udu=\int xdx$$ $$\frac{u^2}{2}=\frac{x^2}{2}+C$$ Substituting $u=f(x)$, we get $f(x{)}^2=x^2+2C$. Using the initial condition $f(0)=1$: $f(0{)}^2=0^2+2C⟹1^2=2C⟹C=\frac{1}{2}$. Thus, the function satisfies $f(x{)}^2=x^2+1$. We need to find $f(1{)}^2$. Substituting $x=1$: $f(1{)}^2=1^2+1=1+1=2$.