Problem 21 - Thomas Calculus in SI Units

Question

Solve the differential equations. $$\frac{1}{x} \frac{d y}{d x}=y e^{x^{2}}+2 \sqrt{y} e^{x^{2}}$$

Step-by-Step Solution

Answer

The solution is \( y = (C' e^{\frac{1}{4} e^{x^2}} - 1)^2 \), with \( C' \) an arbitrary constant.

Step 1: Simplify the Right-Hand Side

The given differential equation is \( \frac{1}{x} \frac{dy}{dx} = y e^{x^2} + 2 \sqrt{y} e^{x^2} \). Factor out \( e^{x^2} \) from the right-hand side: \( \frac{dy}{dx} = e^{x^2} (y + 2 \sqrt{y}) x \).

Step 2: Substitute and Transform

Let \( u = \sqrt{y} \), so \( y = u^2 \) and \( \frac{dy}{dx} = 2u \frac{du}{dx} \). Substitute: \( 2u \frac{du}{dx} = e^{x^2} (u^2 + 2u)x \).

Step 3: Simplify the Transformed Equation

Divide both sides by \( 2u \), to get \( \frac{du}{dx} = \frac{e^{x^2} (u + 2)x}{2} \), i.e. \( \frac{du}{dx} = \frac{e^{x^2} x}{2} + \frac{e^{x^2} ux}{2} \).

Step 4: Separate Variables

Rewrite as \( \frac{du}{1+u} = \frac{e^{x^2} x}{2} dx \). Separate variables: \( \frac{1}{1+u} du = \frac{1}{2} x e^{x^2} dx \).

Step 5: Integrate Both Sides

Integrate the left side: \( \int \frac{1}{1+u} du = \ln|1+u| + C_1 \). For the right side, use substitution method: \( \int \frac{1}{2} x e^{x^2} dx \) let \( v = x^2 \), \( dv = 2x dx \). Thus, \( \int \frac{1}{2} x e^{x^2} dx = \frac{1}{4} \int e^v dv = \frac{1}{4} e^{x^2} + C_2 \).

Step 6: Solve for u

Equating the integrated expressions: \( \ln|1+u| = \frac{1}{4} e^{x^2} + C \). Exponentiate: \( |1+u| = e^{\frac{1}{4} e^{x^2} + C} \). Let \( C' = e^C \), \( 1+u = \pm C' e^{\frac{1}{4} e^{x^2}} \).

Step 7: Solve for y

Recall \( u = \sqrt{y} \), so \( \sqrt{y} = \pm C' e^{\frac{1}{4} e^{x^2}} - 1 \). Squaring both sides gives: \( y = (C' e^{\frac{1}{4} e^{x^2}} - 1)^2 \).

Step 8: Conclusion

Thus, the solution to the differential equation is \( y = (C' e^{\frac{1}{4} e^{x^2}} - 1)^2 \), where \( C' \) is an arbitrary constant.

Key Concepts

Separation of Variables

Separation of Variables is a technique used to solve differential equations where variables can be rearranged and isolated on opposite sides of an equation. This method involves:

• Rewriting the differential equation such that all terms involving one variable are on one side and terms involving the other variable are on the opposite side.
• It leads to an equation where each side can be integrated independently.

In our solution, after substituting and simplifying, the equation was rearranged to: \[ \frac{1}{1+u} du = \frac{1}{2} x e^{x^2} dx \] This form allowed us to separately integrate both sides, ultimately helping separate and solve in terms of the dependent variable and the independent variable. Separation of Variables works well when the equation is easily split into parts dependent on each variable. This approach is foundational for understanding more complicated methods used in solving differential equations.

Integration Techniques

Integration techniques are crucial to solving differential equations after variables have been separated. These techniques involve:

• Recognizing standard integral forms.
• Using substitution to simplify the integral calculation.
• Applying integration by parts when necessary.

In our example:

For the left side, we directly integrated: \[ \int \frac{1}{1+u} du = \ln|1+u| + C_1 \] this standard form leads to a logarithm function.

The right side required substitution, recognizing \( v = x^2 \), yielding: \[ \int \frac{1}{2} x e^{x^2} dx = \frac{1}{4} e^{x^2} + C_2 \]

This highlights the importance of choosing a suitable substitution that simplifies the problem into an easily computable form.

Substitution Method

The substitution method in solving differential equations is a smart way to simplify complex expressions, particularly during integration. This method typically involves:

• Selecting a substitution that transforms complicated functions into simpler ones.
• Changing variables to new ones that make integration straightforward.

In our specific example, substituting \( u = \sqrt{y} \) helped in simplifying the differential equation by reducing it from being directly expressed in terms of \( y \) to \( u \). Additionally, while solving the integral \( \int \frac{1}{2} x e^{x^2} dx \), substitution \( v = x^2 \) transformed the expression into \( \int \frac{1}{4} e^v dv \), which is straightforward to evaluate. This approach makes dealing with complicated integrals manageable, emphasizing how substitution connects different calculus concepts effectively.