Use Lagrange method to write the general solution of the following partial differentialequations.

Use Lagrange method to write the general solution of the following partial differentialequations.(a) x(y2 − u2)ux − y(u2 + x2)uy = u(x2 + y2).(b) (y − ux)ux + (x + yu)uy = x2 + y2.(c) x(x2 + 3y2)ux − y(3x2 + y2)uy = 2u(y2 − x2).(d) (1 + y)ux + (1 + x)uy = u

Find the general solution of the following differential equations

Using the method of separation of variables, solve the following partial differentialequations:(i) 𝜕𝑢𝜕𝑥 = 2 𝜕𝑢𝜕𝑡 + 𝑢; given that 𝑢(𝑥, 0) = 6𝑒−3𝑥.(ii) 4 𝜕𝑢𝜕𝑡 + 𝜕𝑢𝜕𝑥 = 3𝑢; given that 𝑢(𝑥, 0) = 3𝑒−𝑥 − 𝑒−5𝑥

Obtain the general solution of the followingfirst order ODE :G ( ) F ,d t tdt =where F is a constant

To solve the utility maximization problem given the utility function \( u(x, y) = 5xy \), the budget constraint \( 5x + y = 30 \), and non-negative consumption of goods \( x \) and \( y \), follow these steps: 1. **Set up the Lagrangian**: \[ \mathcal{L}(x, y, \lambda) = 5xy + \lambda (30 - 5x - y) \] 2. **Find the partial derivatives and set them to zero**: \[ \frac{\partial \mathcal{L}}{\partial x} = 5y - 5\lambda = 0 \quad \Rightarrow \quad y = \lambda \] \[ \frac{\partial \mathcal{L}}{\partial y} = 5x - \lambda = 0 \quad \Rightarrow \quad \lambda = 5x \] \[ \frac{\partial \mathcal{L}}{\partial \lambda} = 30 - 5x - y = 0 \quad \Rightarrow \quad y = 30 - 5x \] 3. **Substitute \( \lambda \) from the first equation into the second equation**: \[ y = 5x \] 4. **Substitute \( y = 5x \) into the budget constraint**: \[ 5x + 5x = 30 \] \[ 10x = 30 \] \[ x = 3 \] 5. **Find the corresponding \( y \)**: \[ y = 5x = 5 \times 3 = 15 \] So, the x-coordinate of the point that solves this individual's utility maximization problem is \( x = 3 \). The correct answer is: - \( 3 \)