Question

Solve the differential equation. $$ x y+y^{\prime}=100 x $$

Short answer

The solution of the differential equation is: \(y = e^{-1/2x^2} * C + 200 e^{-1/2x^2}*\int{xe^{1/2x^2}} dx\), where \(C\) is an arbitrary constant.

Step-by-step solution

Write in General Form

First, rewrite this into linear differential equation standard form. The original equation is \(x y + y' = 100x\). Subtract \(x y\) from both sides to get \(y' + x y = 100 x\).

Find Integrating Factor

The integrating factor is found by calculating \(e^{\int P(x) dx}\), where \(P(x)\) is the coefficient in front of \(y\) in our equation, in this case \(P(x) = x\). So \(e^{\int x dx} = e ^{1/2x^2}\).

Multiply Original Equation

Multiply every term in the original equation by the calculated integrating factor: \(e^{1/2x^2}y' + x e^{1/2x^2}y =100x e^{1/2x^2}\).

Integration

This equation can now be written as the derivative of a product. So we integrate both sides of this equation: \(\int{d(e^{1/2x^2}y)} = \int{100x e^{1/2x^2} dx}\).

Calculate Result

Solving these integrals and rearranging gives the following solution for y: \(y = e^{-1/2x^2} * C + 200 e^{-1/2x^2}*\int{xe^{1/2x^2}} dx)\), where \(C\) is an arbitrary constant.