Let y_{1}(x) and y_{2}(x) be two solutions of the differential equation \frac{dy}{dx}=x. If y_{1}(0)=0 and y_{2}(0)=4 then what is the number of points of intersection of the curves y_{1}(x) and y_{2}(x)?

Explanation

Integrate \frac{dy}{dx} = x to get y = \frac{x^2}{2} + C. For y_1(0)=0, C=0 \implies y_1 = \frac{x^2}{2}. For y_2(0)=4, C=4 \implies y_2 = \frac{x^2}{2} + 4. Setting them equal gives \frac{x^2}{2} = \frac{x^2}{2} + 4 \implies 0=4, which has no solution. Thus, they never intersect.

Related questions on Calculus

• Let z=[y] and y=[x]-x, where [.] is the greatest integer function. If x is <strong>NOT</strong> an integer but positive, then what i...
• If f(x)=4x+1 and g(x)=kx+2 such that fog(x)=gof(x), then what is the value of k?
• What is the <strong>MINIMUM</strong> value of the function f(x)=\log_{10}(x^{2}+2x+11)?
• What is \int(x^{x})^{2}(1+\ln x)\,dx equal to ?
• What is \int e^{x}\{1+\ln x+x\ln x\}\,dx equal to?