Math Problem Statement
y´=1+x+y+xy
Solution
The given differential equation is:
This can be rewritten as:
This equation is not separable or exact in its current form, but it can be simplified using a substitution technique. One common substitution is to use:
Differentiating both sides with respect to , we get:
Now substitute into the original equation:
Simplifying:
Factor out :
This is a separable equation now. We can rewrite it as:
Integrating both sides:
Exponentiating both sides:
Let , then:
Since , we have:
Finally, solving for :
Thus, the general solution is:
Would you like more details on any step or clarification?
Related Questions:
- How do you solve separable differential equations in general?
- What are the conditions for a differential equation to be exact?
- Can other substitution methods simplify this differential equation further?
- How do boundary or initial conditions affect the constant ?
- What are common techniques to solve first-order differential equations?
Tip:
Always check if a differential equation is separable, exact, or can be transformed via substitution for easier solving methods.
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Math Problem Analysis
Mathematical Concepts
Differential Equations
Substitution Method
Separable Equations
Formulas
y' = 1 + x + y + xy
v = y + 1
\frac{dv}{dx} = v(1 + x)
\ln|v| = \frac{x^2}{2} + x + C
v = C_1 e^{\frac{x^2}{2} + x}
y = C_1 e^{\frac{x^2}{2} + x} - 1
Theorems
Separable Differential Equations
Substitution in Differential Equations
Suitable Grade Level
College Level - Calculus II or Differential Equations
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