Solving a First-Order Differential Equation Using Substitution and Integration Techniques

When faced with the differential equation (frac{dy}{dx} frac{1}{xy - x^2y^3}), it can be effectively simplified and solved through algebraic manipulation and integration. Let's walk through the process step-by-step.

Step 1: Rewrite the Equation

The original differential equation is:

frac{dy}{dx}  frac{1}{xy - x^2y^3}

First, we can factor the denominator for simplicity:

xy - x^2y^3  xy(1 - xy^2)

Thus, we rewrite the differential equation as:

frac{dy}{dx}  frac{1}{xy(1 - xy^2)}

Step 2: Separate Variables

To solve this equation, we can separate the variables by multiplying both sides by (xy(1 - xy^2)) and (dx):

xy(1 - xy^2) , dy  dx

Further isolating (y), we get:

(1 - xy^2) , dy  frac{1}{x} , dx

This step allows us to integrate each side separately.

Step 3: Integrate Both Sides

The left-hand side requires a bit more algebra. We can integrate:

int (1 - xy^2) , dy  int frac{1}{x} , dx

The left-hand side integrates to:

int y , dy - int xy^3 , dy frac{y^2}{2} - frac{x y^4}{4} C_1

The right-hand side integrates to:

int frac{1}{x} , dx ln x C_2

Setting the two integrals equal to each other, we get:

frac{y^2}{2} - frac{x y^4}{4}  ln x   C

To simplify, we can multiply through by 4:

2y^2 - xy^4  4 ln x   C_3

This is the implicit solution to the differential equation. Depending on the context, you might want to solve for (y), but the implicit form is often sufficient.

Step 4: Alternative Solution Using (frac{dx}{dy})

The differential equation can also be solved by considering (frac{dx}{dy}) instead:

frac{dx}{dy}  xy - x^2y^3

This form can be recognized as a Bernoulli equation in terms of (x). Use the substitution (v x^{1-2} x^{-1}), and derive:

frac{dx}{dy}  yx - y^3x^2

Substitute (v) and (frac{dv}{dy}) into the equation:

-1x^2frac{dv}{dy}  yx - y^3x^2

Further manipulation:

frac{dv}{dy}  -yx^{-1} - y^3

frac{dv}{dy}   yx^{-1}  -y^3

This is a linear first-order differential equation. The integrating factor (rho_y) is:

rho_y  e^{int y dy}  e^{frac{y^2}{2}}

Multiplying both sides by the integrating factor:

e^{frac{y^2}{2}} cdot v  -int y^3 e^{frac{y^2}{2}} dy

This integral can be solved using integration by parts:

u  y^2 quad du  2y dy 
 v  e^{frac{y^2}{2}} quad dv  y e^{frac{y^2}{2}} dy

Using integration by parts:

int u dv  u cdot v - int v du 
int y^2 e^{frac{y^2}{2}} dy  y^2 e^{frac{y^2}{2}} - 2 int e^{frac{y^2}{2}} dy

Thus, the solution is:

frac{dv}{dy}   yx^{-1}  -y^3 
int y^3 e^{frac{y^2}{2}} dy  -y^2 e^{frac{y^2}{2}}   2 e^{frac{y^2}{2}} c_1

Finally, solving for (v):(v x^{-1}), we get:

x^{-1}  2 - y^2 e^{-frac{y^2}{2}}   C quad c  -K

The implicit solution is:

1 - 2x - xy^2 e^{frac{y^2}{2}}  Cx

This step-by-step approach ensures a comprehensive solution to the given differential equation.

Summary

The solution to the differential equation can be expressed in either an implicit or explicit form, depending on the requirements of the problem. The key techniques used include variable separation, substitution, and integrating factors.