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The problems in the image are as follows:

### Problem 1:
Using the method of characteristics, solve the following initial value problem:
\[
u_x + u_y = 0, \quad u(x,0) = e^{x^2}
\]

### Problem 2:
Using the method of characteristics, solve the following initial value problem:
\[
u_x + u_y = x^2 + y^2, \quad u(x, y) = 10
\]
where \( x = t \) and \( y = t^2 \), with \( u(0,0) = 0 \).

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Using the method of characteristics, solve the following initial value problems:
1. u_x + u_y = 0, with the initial condition u(x, 0) = e^(x^2).
2. u_x + u_y = x^2 + y^2, with the initial conditions given by (x, y) = (t, t^2) and u(0, 0) = 0.

Learn how to solve initial value problems using the method of characteristics. This guide covers solving PDEs like u_x + u_y = 0 and u_x + u_y = x^2 + y^2 with specific initial conditions. Suitable for undergraduate students studying advanced mathematics.

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