Problem 25 - Essential Calculus

Question

Solve the differential equation. $$ \frac{d y}{d x}=\frac{x+1}{\left(x^{2}+2 x-3\right)^{2}} $$

Step-by-Step Solution

Answer

The general solution of the given differential equation is \(y = 1/4 \ln|x^2 + 2x -3| - (x^2 + 2x -3) / 2 + C_1\).

Step 1: Define the Substitution

Let's define a new variable \(u\) such that \(u = x^2 + 2x - 3\). Then, differentiate \(u\) with respect to \(x\) to get \(du/dx = 2x + 2\). This will allow us to substitute \(u\) into the original equation.

Step 2: Substitute into the Differential Equation

Replace \(x^2+2x-3\) with \(u\) and \(x+1\) with \((du/dx - 2) / 2\) in the original differential equation. This transforms the original equation into: \(dy/dx = (du/dx - 2) / (2u^2)\).

Step 3: Simplify the New Differential Equation

Next, multiply both sides of the equation by \(2u^2\) and rearrange terms to isolate \(dy/dx\), which would result in the equivalent form \(dy = (du - 4u) / (4u^2) dx\). This equation is easier to solve with the separation of variables method.

Step 4: Solve with Separation of Variables and Integration

We then separate the variables and integrate both sides. The left side integrates to \(y+C_1\) with \(C_1\) being the constant of integration. The right side is a standard integral which is \(1/4(\ln |u| - 2u)\). After integrating, we then substitute \(u\) back into the equation to get the solution in terms of original variable.

Step 5: Find the General Solution

Substitute the expression for \(u\) back into our integral to find the general solution of the original differential equation. Thus, we have: \(y = 1/4 \ln|x^2 + 2x -3| - (x^2 + 2x -3) / 2+ C_1\). This is the general solution to the given differential equation.