Solve y'-2xy=x - ScanMath

Question

Solve the differential equation

y = \frac{C e ^{(x^{2})} - 1}{2}, C \in R

Evaluate

y^{'} - 2 x y = x

Move the expression to the right side

y^{'} = x + 2 x y

Rewrite the expression

\frac{d y}{d x} = x + 2 x y

Rewrite the expression

\frac{d y}{d x} = x (1 + 2 y)

Rewrite the expression

\frac{1}{1 + 2 y} \times \frac{d y}{d x} = x (1 + 2 y) \times \frac{1}{1 + 2 y}

Multiply the terms

\frac{1}{1 + 2 y} \times \frac{d y}{d x} = x

Transform the expression

\frac{1}{1 + 2 y} \times d y = x d x

Integrate the left-hand side of the equation with respect to y and the right-hand side of the equation with respect to x

\int \frac{1}{1 + 2 y} d y = \int x d x

Calculate

More Steps

\frac{1}{2} ln (y + \frac{1}{2}) + C_{1} = \int x d x, C_{1} \in R

Calculate

More Steps

\frac{1}{2} ln (y + \frac{1}{2}) + C_{1} = \frac{x ^{2}}{2} + C_{2}, C_{1} \in R, C_{2} \in R

Since the integral constants C_{1} and C_{2} are arbitrary constants, replace them with constant C

\frac{1}{2} ln (y + \frac{1}{2}) = \frac{x ^{2}}{2} + C, C \in R

Calculate

More Steps

y = \frac{2 e ^{x^{2} + C} - 1}{2}, C \in R

Rewrite the expression

More Steps

y = \frac{2 C e ^{(x^{2})} - 1}{2}, C \in R

Solution

y = \frac{C e ^{(x^{2})} - 1}{2}, C \in R

Show Solution