本题目来源于试卷: Differential Equations，类别为 IB数学
[问答题]
Consider the differe/ nv v(l-luyva12kkx8+ta4 wr ob;+ rdmkde:dn6uef59 pw/d+ntial equation
$x^{2} \frac{\mathrm{d} y}{\mathrm{~d} x}+6 x^{2}=y^{2}$
for x>0 and y>3 x . It is given that y=4 when x=1 .
1. Use Euler's method, with a step length of 0.08 , to find an approximate value for y when x=1.4 .
2. Use the substitution y=v x to show that $x \frac{\mathrm{d} v}{\mathrm{~d} x}=v^{2}-v-6 $.
3. By solving the differential equation, show that $y=\frac{18 x+2 x^{6}}{6-x^{5}}$ .
4. 1. Find the actual value of y when x=1.4 .
2. Using the graph of $ y=\frac{18 x+2 x^{6}}{6-x^{5}} $, suggest a reason why the approximation given by Euler's method in part (a) is not a good estimate to the actual value of y at x=1.4
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